{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1a. Determine which quadrant 127.5° is in. Explain how you know."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Angles and Quadrants"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of how angles in standard position are classified into quadrants based on their degree measure."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quadrant: The coordinate plane is divided into four quadrants by the x- and y-axes."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Standard Position: An angle whose vertex is at the origin and whose initial side lies along the positive x-axis."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall the degree ranges for each quadrant:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quadrant I: "},{"type":"inlineMath","attrs":{"latex":"0^\\circ"}},{"type":"text","text":" to "},{"type":"inlineMath","attrs":{"latex":"90^\\circ"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quadrant II: "},{"type":"inlineMath","attrs":{"latex":"90^\\circ"}},{"type":"text","text":" to "},{"type":"inlineMath","attrs":{"latex":"180^\\circ"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quadrant III: "},{"type":"inlineMath","attrs":{"latex":"180^\\circ"}},{"type":"text","text":" to "},{"type":"inlineMath","attrs":{"latex":"270^\\circ"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quadrant IV: "},{"type":"inlineMath","attrs":{"latex":"270^\\circ"}},{"type":"text","text":" to "},{"type":"inlineMath","attrs":{"latex":"360^\\circ"}}]}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compare "},{"type":"inlineMath","attrs":{"latex":"127.5^\\circ"}},{"type":"text","text":" to these ranges to determine which quadrant it falls into."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Explain your reasoning based on the comparison above."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"127.5^\\circ"}},{"type":"text","text":" is in Quadrant II because it is greater than "},{"type":"inlineMath","attrs":{"latex":"90^\\circ"}},{"type":"text","text":" but less than "},{"type":"inlineMath","attrs":{"latex":"180^\\circ"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"We know this because Quadrant II contains all angles between "},{"type":"inlineMath","attrs":{"latex":"90^\\circ"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"180^\\circ"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1b. Draw the angle in its appropriate quadrant below."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Drawing Angles in Standard Position"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question checks your ability to represent an angle in standard position on the coordinate plane."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Initial Side: The starting position of the angle (along the positive x-axis)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Terminal Side: The position after rotating the initial side by the given angle."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Start by drawing the x- and y-axes."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Draw the initial side of the angle along the positive x-axis."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Measure "},{"type":"inlineMath","attrs":{"latex":"127.5^\\circ"}},{"type":"text","text":" counterclockwise from the positive x-axis."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Mark the terminal side in Quadrant II, as determined in part (a)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The angle "},{"type":"inlineMath","attrs":{"latex":"127.5^\\circ"}},{"type":"text","text":" is drawn starting from the positive x-axis, rotating counterclockwise, and its terminal side lands in Quadrant II."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1c. In your picture above, draw the right triangle in the appropriate quadrant. Clearly label the lengths of the sides correctly and label the angles (i.e. the reference angle and its complement) in the triangle correctly as well."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Reference Angles and Right Triangles in the Coordinate Plane"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to construct a right triangle corresponding to a given angle in standard position and to identify the reference angle."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Reference Angle: The acute angle formed by the terminal side of the given angle and the x-axis."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Complement: The other acute angle in the right triangle (since the sum of the two non-right angles in a right triangle is "},{"type":"inlineMath","attrs":{"latex":"90^\\circ"}},{"type":"text","text":")."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"From your drawing in part (b), drop a perpendicular from the terminal side of the angle to the x-axis to form a right triangle."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Label the right angle at the intersection with the x-axis."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the reference angle: "},{"type":"inlineMath","attrs":{"latex":"\\text{Reference angle} = 180^\\circ - 127.5^\\circ"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Label the reference angle and its complement inside the triangle."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Assign side lengths based on the triangle's orientation in Quadrant II (signs of x and y values)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The right triangle is drawn in Quadrant II, with the reference angle labeled as "},{"type":"inlineMath","attrs":{"latex":"52.5^\\circ"}},{"type":"text","text":" (since "},{"type":"inlineMath","attrs":{"latex":"180^\\circ - 127.5^\\circ = 52.5^\\circ"}},{"type":"text","text":"). The complement is "},{"type":"inlineMath","attrs":{"latex":"37.5^\\circ"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The side adjacent to the reference angle (x-direction) is negative, and the side opposite (y-direction) is positive."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1d. Use your picture above to find the following ratios: sec(127.5°), tan(127.5°)"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Trigonometric Ratios in Non-Quadrant I Angles"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the reference triangle to find trigonometric ratios for angles outside Quadrant I, considering the signs of the sides."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sec(\\theta) = \\frac{\\text{hypotenuse}}{\\text{adjacent}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\tan(\\theta) = \\frac{\\text{opposite}}{\\text{adjacent}}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the side lengths from your triangle in part (c), making sure to use the correct signs for Quadrant II (adjacent is negative, opposite is positive)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the ratio for "},{"type":"inlineMath","attrs":{"latex":"\\sec(127.5^\\circ)"}},{"type":"text","text":" using the formula above."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the ratio for "},{"type":"inlineMath","attrs":{"latex":"\\tan(127.5^\\circ)"}},{"type":"text","text":" using the formula above."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Do not compute the final values yet; just write the expressions using the side lengths."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sec(127.5^\\circ) = -\\frac{\\text{hypotenuse}}{\\text{adjacent}}"}},{"type":"text","text":" (using the negative adjacent side in Quadrant II)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\tan(127.5^\\circ) = -\\frac{\\text{opposite}}{\\text{adjacent}}"}},{"type":"text","text":" (opposite is positive, adjacent is negative in Quadrant II)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Plug in the specific side lengths from your triangle to get the exact values."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1e. Determine which quadrant −37.5° is in. Explain how you know."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Negative Angles and Quadrants"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of how negative angles are measured and how to determine their quadrant."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Negative Angle: Measured clockwise from the positive x-axis."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that negative angles are measured clockwise from the positive x-axis."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Determine the equivalent positive angle by adding "},{"type":"inlineMath","attrs":{"latex":"360^\\circ"}},{"type":"text","text":" if needed: "},{"type":"inlineMath","attrs":{"latex":"-37.5^\\circ + 360^\\circ"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compare the resulting angle to the quadrant ranges to determine the correct quadrant."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"-37.5^\\circ"}},{"type":"text","text":" is in Quadrant IV because it is a small negative angle, measured clockwise from the positive x-axis, landing just below the x-axis."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1h. Determine which quadrant −150° is in. Explain how you know."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Negative Angles and Quadrants"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to determine the quadrant of a negative angle."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Negative Angle: Measured clockwise from the positive x-axis."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add "},{"type":"inlineMath","attrs":{"latex":"360^\\circ"}},{"type":"text","text":" to "},{"type":"inlineMath","attrs":{"latex":"-150^\\circ"}},{"type":"text","text":" to find the coterminal positive angle: "},{"type":"inlineMath","attrs":{"latex":"-150^\\circ + 360^\\circ"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compare the resulting angle to the quadrant ranges."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"State which quadrant the angle is in and explain your reasoning."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"-150^\\circ + 360^\\circ = 210^\\circ"}},{"type":"text","text":", which is in Quadrant III (between "},{"type":"inlineMath","attrs":{"latex":"180^\\circ"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"270^\\circ"}},{"type":"text","text":")."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1I. Convert the angle −150° into radian measure."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Degree-Radian Conversion"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to convert between degrees and radians."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Formula:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\text{Radians} = \\text{Degrees} \\times \\frac{\\pi}{180^\\circ}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the conversion formula: "},{"type":"inlineMath","attrs":{"latex":"\\text{Radians} = \\text{Degrees} \\times \\frac{\\pi}{180^\\circ}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Plug in "},{"type":"inlineMath","attrs":{"latex":"-150^\\circ"}},{"type":"text","text":" for degrees."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the fraction, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"-150^\\circ = -150 \\times \\frac{\\pi}{180} = -\\frac{5\\pi}{6}"}},{"type":"text","text":" radians."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1k. Use your picture above to find the following ratios: sin(−150°), sec(−150°)"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Trigonometric Ratios for Angles in Quadrant III"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use a reference triangle to find trigonometric ratios for an angle in Quadrant III, considering the signs of the sides."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin(\\theta) = \\frac{\\text{opposite}}{\\text{hypotenuse}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sec(\\theta) = \\frac{\\text{hypotenuse}}{\\text{adjacent}}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Draw the reference triangle for "},{"type":"inlineMath","attrs":{"latex":"-150^\\circ"}},{"type":"text","text":" in Quadrant III, as in previous steps."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Label the side lengths, making sure to use the correct signs for Quadrant III (both x and y are negative)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the ratio for "},{"type":"inlineMath","attrs":{"latex":"\\sin(-150^\\circ)"}},{"type":"text","text":" using the formula above."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the ratio for "},{"type":"inlineMath","attrs":{"latex":"\\sec(-150^\\circ)"}},{"type":"text","text":" using the formula above."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Do not compute the final values yet; just write the expressions using the side lengths."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin(-150^\\circ) = -\\frac{1}{2}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sec(-150^\\circ) = -\\frac{2}{\\sqrt{3}}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Both are negative because in Quadrant III, both x and y are negative."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2. Find the six trigonometric ratios of θ in the triangle below (with sides 7, 24, 25)."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Trigonometric Ratios from Right Triangles"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find all six trigonometric ratios (sine, cosine, tangent, secant, cosecant, cotangent) for a given angle in a right triangle."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin \\theta = \\frac{\\text{opposite}}{\\text{hypotenuse}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos \\theta = \\frac{\\text{adjacent}}{\\text{hypotenuse}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\tan \\theta = \\frac{\\text{opposite}}{\\text{adjacent}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\csc \\theta = \\frac{\\text{hypotenuse}}{\\text{opposite}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sec \\theta = \\frac{\\text{hypotenuse}}{\\text{adjacent}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cot \\theta = \\frac{\\text{adjacent}}{\\text{opposite}}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the sides of the triangle: opposite, adjacent, and hypotenuse (7, 24, 25)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the expressions for each trigonometric ratio using the side lengths."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Do not simplify or compute the final values yet; just set up the ratios."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin \\theta = \\frac{7}{25}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos \\theta = \\frac{24}{25}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\tan \\theta = \\frac{7}{24}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\csc \\theta = \\frac{25}{7}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sec \\theta = \\frac{25}{24}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cot \\theta = \\frac{24}{7}"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. Suppose θ is in quadrant III and cos(θ) = −3/5. Draw θ in the correct standard position and then draw the right triangle corresponding to the correct reference angle θ' in that quadrant. Label θ and θ' in your picture. Label the signed lengths of each side of the right triangle in the picture. Find the other 5 trigonometric ratios of θ from your picture."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Trigonometric Ratios in Quadrant III"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use a given trigonometric value and quadrant information to construct a reference triangle and find all other trigonometric ratios."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos \\theta = \\frac{\\text{adjacent}}{\\text{hypotenuse}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the Pythagorean Theorem to find the missing side: "},{"type":"inlineMath","attrs":{"latex":"a^2 + b^2 = c^2"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Remember the signs of the sides in Quadrant III (both x and y are negative)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Since "},{"type":"inlineMath","attrs":{"latex":"\\cos \\theta = -\\frac{3}{5}"}},{"type":"text","text":", assign the adjacent side as "},{"type":"inlineMath","attrs":{"latex":"-3"}},{"type":"text","text":" and the hypotenuse as $5$ (hypotenuse is always positive)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the Pythagorean Theorem to find the length of the opposite side."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Assign the correct sign to the opposite side (negative in Quadrant III)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the expressions for "},{"type":"inlineMath","attrs":{"latex":"\\sin \\theta"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\tan \\theta"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\csc \\theta"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\sec \\theta"}},{"type":"text","text":", and "},{"type":"inlineMath","attrs":{"latex":"\\cot \\theta"}},{"type":"text","text":" using the side lengths."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Do not compute the final values yet; just set up the ratios."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Opposite side: "},{"type":"inlineMath","attrs":{"latex":"-4"}},{"type":"text","text":" (since "},{"type":"inlineMath","attrs":{"latex":"(-3)^2 + (-4)^2 = 9 + 16 = 25"}},{"type":"text","text":")"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin \\theta = -\\frac{4}{5}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\tan \\theta = \\frac{-4}{-3} = \\frac{4}{3}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\csc \\theta = -\\frac{5}{4}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sec \\theta = -\\frac{5}{3}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cot \\theta = \\frac{-3}{-4} = \\frac{3}{4}"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4. Find the domains of the functions "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\frac{x}{x^2 - x - 6}"}},{"type":"text","marks":[{"type":"bold"}],"text":" and "},{"type":"inlineMath","attrs":{"latex":"g(x) = \\frac{1}{\\sqrt{x^2 - x - 6}}"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Domain of Rational and Radical Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the domain of functions involving denominators and square roots."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Concepts:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For rational functions, the denominator cannot be zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For square root functions, the radicand must be non-negative, and the denominator cannot be zero."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":", set the denominator "},{"type":"inlineMath","attrs":{"latex":"x^2 - x - 6 \\neq 0"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For "},{"type":"inlineMath","attrs":{"latex":"g(x)"}},{"type":"text","text":", set "},{"type":"inlineMath","attrs":{"latex":"x^2 - x - 6 > 0"}},{"type":"text","text":" (since the denominator must be positive and not zero)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve the inequalities to find the intervals for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express the domains in interval notation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":": Domain is all real numbers except "},{"type":"inlineMath","attrs":{"latex":"x = 3"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x = -2"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For "},{"type":"inlineMath","attrs":{"latex":"g(x)"}},{"type":"text","text":": Domain is "},{"type":"inlineMath","attrs":{"latex":"(-\\infty, -2) \\cup (3, \\infty)"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5a. Suppose "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\sqrt{x^2}"}},{"type":"text","marks":[{"type":"bold"}],"text":" and "},{"type":"inlineMath","attrs":{"latex":"g(x) = x^2 - 100x"}},{"type":"text","marks":[{"type":"bold"}],"text":". Find "},{"type":"inlineMath","attrs":{"latex":"(f \\circ g)(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Function Composition"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to compose two functions, meaning to substitute one function into another."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Formula:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"(f \\circ g)(x) = f(g(x))"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write "},{"type":"inlineMath","attrs":{"latex":"g(x) = x^2 - 100x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute "},{"type":"inlineMath","attrs":{"latex":"g(x)"}},{"type":"text","text":" into "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"f(g(x)) = \\sqrt{(g(x))^2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the expression for "},{"type":"inlineMath","attrs":{"latex":"(f \\circ g)(x)"}},{"type":"text","text":" in terms of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Do not simplify the final expression yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"(f \\circ g)(x) = \\sqrt{(x^2 - 100x)^2} = |x^2 - 100x|"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5b. Find the domain of "},{"type":"inlineMath","attrs":{"latex":"(f \\circ g)(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":" from above."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Domain of Composed Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to determine the domain of a function composition, considering the domains of both functions involved."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Concepts:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The domain of "},{"type":"inlineMath","attrs":{"latex":"f \\circ g"}},{"type":"text","text":" is all "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" such that $x$ is in the domain of "},{"type":"inlineMath","attrs":{"latex":"g"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"g(x)"}},{"type":"text","text":" is in the domain of "},{"type":"inlineMath","attrs":{"latex":"f"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\sqrt{x^2}"}},{"type":"text","text":", the domain is all real numbers (since "},{"type":"inlineMath","attrs":{"latex":"x^2"}},{"type":"text","text":" is always non-negative)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Determine the domain of "},{"type":"inlineMath","attrs":{"latex":"g(x)"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"x^2 - 100x"}},{"type":"text","text":" is defined for all real "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check if "},{"type":"inlineMath","attrs":{"latex":"f(g(x))"}},{"type":"text","text":" is defined for all "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" in the domain of "},{"type":"inlineMath","attrs":{"latex":"g"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Conclude the domain for "},{"type":"inlineMath","attrs":{"latex":"(f \\circ g)(x)"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The domain of "},{"type":"inlineMath","attrs":{"latex":"(f \\circ g)(x)"}},{"type":"text","text":" is all real numbers, "},{"type":"inlineMath","attrs":{"latex":"(-\\infty, \\infty)"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6. For the following statement, fill in the blank with one of the words ALL, SOME, or NONE. If all or no angles satisfy the condition explain why all or none of them satisfy the property. If some angles satisfy the property, then give an example of an angle that satisfies the property and an angle that does not satisfy the property."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"_____ of the acute angles A with "},{"type":"inlineMath","attrs":{"latex":"0^\\circ < A < 90^\\circ"}},{"type":"text","text":" have a tangent ratio that is bigger than 1: i.e. "},{"type":"inlineMath","attrs":{"latex":"\\tan A > 1"}},{"type":"text","text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Trigonometric Ratios for Acute Angles"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the range of the tangent function for acute angles."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Concepts:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For "},{"type":"inlineMath","attrs":{"latex":"0^\\circ < A < 90^\\circ"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\tan A"}},{"type":"text","text":" increases from 0 to "},{"type":"inlineMath","attrs":{"latex":"+\\infty"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\tan A = 1"}},{"type":"text","text":" when "},{"type":"inlineMath","attrs":{"latex":"A = 45^\\circ"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Consider the values of "},{"type":"inlineMath","attrs":{"latex":"A"}},{"type":"text","text":" for which "},{"type":"inlineMath","attrs":{"latex":"\\tan A > 1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Determine if this is true for all, some, or none of the acute angles."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Provide an example of an angle that satisfies "},{"type":"inlineMath","attrs":{"latex":"\\tan A > 1"}},{"type":"text","text":" and one that does not."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"SOME of the acute angles have "},{"type":"inlineMath","attrs":{"latex":"\\tan A > 1"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For example, "},{"type":"inlineMath","attrs":{"latex":"A = 60^\\circ"}},{"type":"text","text":" has "},{"type":"inlineMath","attrs":{"latex":"\\tan 60^\\circ = \\sqrt{3} > 1"}},{"type":"text","text":", but "},{"type":"inlineMath","attrs":{"latex":"A = 30^\\circ"}},{"type":"text","text":" has "},{"type":"inlineMath","attrs":{"latex":"\\tan 30^\\circ = \\frac{1}{\\sqrt{3}} < 1"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7. For the following statement, fill in the blank with one of the words ALL, SOME, or NONE. If all or none of the functions satisfy the property, then explain why they all or none at all satisfy the property. If some of the functions satisfy the property, then show an example of a function that satisfies the property and an example that does not satisfy it."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"_____ of the continuous functions "},{"type":"inlineMath","attrs":{"latex":"y = f(x)"}},{"type":"text","text":" that are increasing on an interval "},{"type":"inlineMath","attrs":{"latex":"[a, b]"}},{"type":"text","text":" have an average rate of change that is positive on that interval."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Average Rate of Change and Increasing Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the relationship between increasing functions and the sign of their average rate of change."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Concepts:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"If a function is increasing on "},{"type":"inlineMath","attrs":{"latex":"[a, b]"}},{"type":"text","text":", then "},{"type":"inlineMath","attrs":{"latex":"f(b) > f(a)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The average rate of change is "},{"type":"inlineMath","attrs":{"latex":"\\frac{f(b) - f(a)}{b - a}"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall the definition of an increasing function on "},{"type":"inlineMath","attrs":{"latex":"[a, b]"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Analyze the formula for average rate of change and its sign when "},{"type":"inlineMath","attrs":{"latex":"f(b) > f(a)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"b > a"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Decide if this property holds for all, some, or none of such functions."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"ALL of the continuous functions that are increasing on "},{"type":"inlineMath","attrs":{"latex":"[a, b]"}},{"type":"text","text":" have a positive average rate of change on that interval, because "},{"type":"inlineMath","attrs":{"latex":"f(b) > f(a)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"b > a"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8a. Consider the polynomial function "},{"type":"inlineMath","attrs":{"latex":"p(x) = 3(x + 2)^2(x + 4)^3(x - 5)(x - 7)"}},{"type":"text","marks":[{"type":"bold"}],"text":". Find the roots of "},{"type":"inlineMath","attrs":{"latex":"p"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Roots of Polynomial Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the zeros (roots) of a polynomial by setting the function equal to zero and solving for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Concepts:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Roots are values of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" that make "},{"type":"inlineMath","attrs":{"latex":"p(x) = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set each factor equal to zero and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set "},{"type":"inlineMath","attrs":{"latex":"p(x) = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set each factor equal to zero: "},{"type":"inlineMath","attrs":{"latex":"(x + 2)^2 = 0"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"(x + 4)^3 = 0"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"(x - 5) = 0"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"(x - 7) = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve each equation for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to find the roots."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The roots are "},{"type":"inlineMath","attrs":{"latex":"x = -2"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x = -4"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x = 5"}},{"type":"text","text":", and "},{"type":"inlineMath","attrs":{"latex":"x = 7"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8b. State the multiplicities of each of the roots."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Multiplicity of Roots"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to identify the multiplicity of each root from the exponents in the factored form of a polynomial."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Concepts:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The multiplicity of a root is the exponent of the corresponding factor."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Look at the exponents of each factor in "},{"type":"inlineMath","attrs":{"latex":"p(x)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Assign the multiplicity to each root based on the exponent."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x = -2"}},{"type":"text","text":" has multiplicity 2, "},{"type":"inlineMath","attrs":{"latex":"x = -4"}},{"type":"text","text":" has multiplicity 3, "},{"type":"inlineMath","attrs":{"latex":"x = 5"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x = 7"}},{"type":"text","text":" each have multiplicity 1."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8c. Determine the degree of the polynomial."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Degree of a Polynomial"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the degree of a polynomial by adding the exponents of all the factors."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Concepts:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The degree is the sum of the exponents of all the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" factors in the expanded form."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add the exponents: $2"},{"type":"inlineMath","attrs":{"latex":" (from "}},{"type":"text","text":"(x + 2)^2"},{"type":"inlineMath","attrs":{"latex":"), $3"}},{"type":"text","text":" (from "},{"type":"inlineMath","attrs":{"latex":"(x + 4)^3"}},{"type":"text","text":"), $1$ (from $(x - 5)"},{"type":"inlineMath","attrs":{"latex":"), and $1"}},{"type":"text","text":" (from "},{"type":"inlineMath","attrs":{"latex":"(x - 7)"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sum these values to find the degree."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The degree of the polynomial is "},{"type":"inlineMath","attrs":{"latex":"2 + 3 + 1 + 1 = 7"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8d. Determine the end behavior of the polynomial."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: End Behavior of Polynomials"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to determine how a polynomial behaves as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" approaches "},{"type":"inlineMath","attrs":{"latex":"\\pm\\infty"}},{"type":"text","text":", based on its degree and leading coefficient."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Concepts:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For degree "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" and leading coefficient "},{"type":"inlineMath","attrs":{"latex":"a_n"}},{"type":"text","text":":"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"If "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" is odd and "},{"type":"inlineMath","attrs":{"latex":"a_n > 0"}},{"type":"text","text":", as "},{"type":"inlineMath","attrs":{"latex":"x \\to -\\infty"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"p(x) \\to -\\infty"}},{"type":"text","text":"; as "},{"type":"inlineMath","attrs":{"latex":"x \\to \\infty"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"p(x) \\to \\infty"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"If "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" is even and "},{"type":"inlineMath","attrs":{"latex":"a_n > 0"}},{"type":"text","text":", both ends go up."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the degree (from part c) and the leading coefficient (from the constant in front)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Determine if the degree is odd or even."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the rules above to describe the end behavior as "},{"type":"inlineMath","attrs":{"latex":"x \\to \\infty"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x \\to -\\infty"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Since the degree is 7 (odd) and the leading coefficient is positive (3), as "},{"type":"inlineMath","attrs":{"latex":"x \\to -\\infty"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"p(x) \\to -\\infty"}},{"type":"text","text":"; as "},{"type":"inlineMath","attrs":{"latex":"x \\to \\infty"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"p(x) \\to \\infty"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q9(1). For "},{"type":"inlineMath","attrs":{"latex":"f(x) = -2x^2 - 3x + 4"}},{"type":"text","marks":[{"type":"bold"}],"text":", a) find "},{"type":"inlineMath","attrs":{"latex":"f(x + h)"}},{"type":"text","marks":[{"type":"bold"}],"text":", b) compute "},{"type":"inlineMath","attrs":{"latex":"f(x + h) - f(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":", c) compute the difference quotient "},{"type":"inlineMath","attrs":{"latex":"\\frac{f(x +