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Find all solutions of the equation between 0 and 2π: 2 sin x − √3 = 0"}]},{"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: Solving Trigonometric Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve basic trigonometric equations for all solutions within a given interval."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"sin x: Sine function"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To solve equations like "},{"type":"inlineMath","attrs":{"latex":"a \\sin x + b = 0"}},{"type":"text","text":", isolate "},{"type":"inlineMath","attrs":{"latex":"\\sin x"}},{"type":"text","text":" and use the unit circle to find all solutions in the interval."}]}]}]},{"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 isolating "},{"type":"inlineMath","attrs":{"latex":"\\sin x"}},{"type":"text","text":" in the equation: "},{"type":"inlineMath","attrs":{"latex":"2 \\sin x - \\sqrt{3} = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{3}"}},{"type":"text","text":" to both sides to get "},{"type":"inlineMath","attrs":{"latex":"2 \\sin x = \\sqrt{3}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide both sides by 2: "},{"type":"inlineMath","attrs":{"latex":"\\sin x = \\frac{\\sqrt{3}}{2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall the values of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" in "},{"type":"inlineMath","attrs":{"latex":"[0, 2\\pi]"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"\\sin x = \\frac{\\sqrt{3}}{2}"}},{"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 solutions are "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{\\pi}{3}"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{2\\pi}{3}"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"These are the angles in "},{"type":"inlineMath","attrs":{"latex":"[0, 2\\pi]"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"\\sin x = \\frac{\\sqrt{3}}{2}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2. Find all solutions of the equation between 0 and 2π: sec x + √2 = 0"}]},{"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: Solving Trigonometric Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the secant function and how to solve equations involving it."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"sec x: Secant function, "},{"type":"inlineMath","attrs":{"latex":"\\sec x = \\frac{1}{\\cos x}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To solve "},{"type":"inlineMath","attrs":{"latex":"\\sec x + a = 0"}},{"type":"text","text":", isolate "},{"type":"inlineMath","attrs":{"latex":"\\sec x"}},{"type":"text","text":" and rewrite in terms of "},{"type":"inlineMath","attrs":{"latex":"\\cos 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":"Isolate "},{"type":"inlineMath","attrs":{"latex":"\\sec x"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"\\sec x + \\sqrt{2} = 0 \\implies \\sec x = -\\sqrt{2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that "},{"type":"inlineMath","attrs":{"latex":"\\sec x = \\frac{1}{\\cos x}"}},{"type":"text","text":", so "},{"type":"inlineMath","attrs":{"latex":"\\frac{1}{\\cos x} = -\\sqrt{2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":"\\cos x"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"\\cos x = -\\frac{1}{\\sqrt{2}}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rationalize the denominator if needed, and recall the angles in "},{"type":"inlineMath","attrs":{"latex":"[0, 2\\pi]"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"\\cos x"}},{"type":"text","text":" has this value."}]}]}]},{"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 solutions are "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{3\\pi}{4}"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{5\\pi}{4}"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"These are the angles in "},{"type":"inlineMath","attrs":{"latex":"[0, 2\\pi]"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"\\cos x = -\\frac{\\sqrt{2}}{2}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. Find all solutions of the equation between 0 and 2π: √3 cot x − 1 = 0"}]},{"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: Solving Trigonometric Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve equations involving the cotangent function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"cot x: Cotangent function, "},{"type":"inlineMath","attrs":{"latex":"\\cot x = \\frac{1}{\\tan x}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To solve "},{"type":"inlineMath","attrs":{"latex":"a \\cot x + b = 0"}},{"type":"text","text":", isolate "},{"type":"inlineMath","attrs":{"latex":"\\cot x"}},{"type":"text","text":" and use the unit circle."}]}]}]},{"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":"Isolate "},{"type":"inlineMath","attrs":{"latex":"\\cot x"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{3} \\cot x - 1 = 0 \\implies \\sqrt{3} \\cot x = 1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide both sides by "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{3}"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"\\cot x = \\frac{1}{\\sqrt{3}}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that "},{"type":"inlineMath","attrs":{"latex":"\\cot x = \\frac{1}{\\tan x}"}},{"type":"text","text":", so "},{"type":"inlineMath","attrs":{"latex":"\\tan x = \\sqrt{3}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find all "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" in "},{"type":"inlineMath","attrs":{"latex":"[0, 2\\pi]"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"\\tan x = \\sqrt{3}"}},{"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 solutions are "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{\\pi}{3}"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{4\\pi}{3}"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"These are the angles in "},{"type":"inlineMath","attrs":{"latex":"[0, 2\\pi]"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"\\tan x = \\sqrt{3}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4. Find all solutions of the equation between 0 and 2π: √3 sin x + √3 = 0"}]},{"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: Solving Trigonometric Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve equations involving the sine function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"sin x: Sine function"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To solve "},{"type":"inlineMath","attrs":{"latex":"a \\sin x + b = 0"}},{"type":"text","text":", isolate "},{"type":"inlineMath","attrs":{"latex":"\\sin x"}},{"type":"text","text":" and use the unit circle."}]}]}]},{"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":"Isolate "},{"type":"inlineMath","attrs":{"latex":"\\sin x"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{3} \\sin x + \\sqrt{3} = 0 \\implies \\sqrt{3} \\sin x = -\\sqrt{3}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide both sides by "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{3}"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"\\sin x = -1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall the values of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" in "},{"type":"inlineMath","attrs":{"latex":"[0, 2\\pi]"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"\\sin x = -1"}},{"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 solution is "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{3\\pi}{2}"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This is the only value in "},{"type":"inlineMath","attrs":{"latex":"[0, 2\\pi]"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"\\sin x = -1"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5. Find all solutions of the equation between 0 and 2π: √2 csc x − 1 = 0"}]},{"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: Solving Trigonometric Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the cosecant function and how to solve equations involving it."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"csc x: Cosecant function, "},{"type":"inlineMath","attrs":{"latex":"\\csc x = \\frac{1}{\\sin x}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To solve "},{"type":"inlineMath","attrs":{"latex":"a \\csc x + b = 0"}},{"type":"text","text":", isolate "},{"type":"inlineMath","attrs":{"latex":"\\csc x"}},{"type":"text","text":" and rewrite in terms of "},{"type":"inlineMath","attrs":{"latex":"\\sin 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":"Isolate "},{"type":"inlineMath","attrs":{"latex":"\\csc x"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{2} \\csc x - 1 = 0 \\implies \\sqrt{2} \\csc x = 1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide both sides by "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{2}"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"\\csc x = \\frac{1}{\\sqrt{2}}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that "},{"type":"inlineMath","attrs":{"latex":"\\csc x = \\frac{1}{\\sin x}"}},{"type":"text","text":", so "},{"type":"inlineMath","attrs":{"latex":"\\sin x = \\sqrt{2}"}},{"type":"text","text":" (but check if this is possible)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Consider the possible values of "},{"type":"inlineMath","attrs":{"latex":"\\sin x"}},{"type":"text","text":" and whether "},{"type":"inlineMath","attrs":{"latex":"\\sin x = \\sqrt{2}"}},{"type":"text","text":" is valid."}]}]}]},{"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":"There are no solutions, because "},{"type":"inlineMath","attrs":{"latex":"\\sin x"}},{"type":"text","text":" cannot be greater than 1 or less than -1, and "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{2} > 1"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6. Find all solutions of the equation between 0° and 360°: 2 cos 2θ + √3 = 0"}]},{"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: Solving Trigonometric Equations with Double Angles"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve equations involving double angles and to find all solutions in degrees."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"cos 2θ: Double angle formula for cosine"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To solve "},{"type":"inlineMath","attrs":{"latex":"a \\cos 2\\theta + b = 0"}},{"type":"text","text":", isolate "},{"type":"inlineMath","attrs":{"latex":"\\cos 2\\theta"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"\\theta"}},{"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":"Isolate "},{"type":"inlineMath","attrs":{"latex":"\\cos 2\\theta"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"2 \\cos 2\\theta + \\sqrt{3} = 0 \\implies 2 \\cos 2\\theta = -\\sqrt{3}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide both sides by 2: "},{"type":"inlineMath","attrs":{"latex":"\\cos 2\\theta = -\\frac{\\sqrt{3}}{2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find all "},{"type":"inlineMath","attrs":{"latex":"2\\theta"}},{"type":"text","text":" in "},{"type":"inlineMath","attrs":{"latex":"[0°, 360°]"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"\\cos 2\\theta = -\\frac{\\sqrt{3}}{2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For each solution for "},{"type":"inlineMath","attrs":{"latex":"2\\theta"}},{"type":"text","text":", solve for "},{"type":"inlineMath","attrs":{"latex":"\\theta"}},{"type":"text","text":" in "},{"type":"inlineMath","attrs":{"latex":"[0°, 360°]"}},{"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 solutions are "},{"type":"inlineMath","attrs":{"latex":"\\theta = 105°"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"\\theta = 255°"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"These come from "},{"type":"inlineMath","attrs":{"latex":"2\\theta = 210°"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"2\\theta = 330°"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7. Find all solutions of the equation between 0 and 2π: 5 tan x − 5√3 = 0"}]},{"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: Solving Trigonometric Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve equations involving the tangent function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"tan x: Tangent function"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To solve "},{"type":"inlineMath","attrs":{"latex":"a \\tan x + b = 0"}},{"type":"text","text":", isolate "},{"type":"inlineMath","attrs":{"latex":"\\tan x"}},{"type":"text","text":" and use the unit circle."}]}]}]},{"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":"Isolate "},{"type":"inlineMath","attrs":{"latex":"\\tan x"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"5 \\tan x - 5\\sqrt{3} = 0 \\implies 5 \\tan x = 5\\sqrt{3}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide both sides by 5: "},{"type":"inlineMath","attrs":{"latex":"\\tan x = \\sqrt{3}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall the values of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" in "},{"type":"inlineMath","attrs":{"latex":"[0, 2\\pi]"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"\\tan x = \\sqrt{3}"}},{"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 solutions are "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{\\pi}{3}"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{4\\pi}{3}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8. Find all solutions of the equation between 0 and 2π: 2 cos 2x − 1 = 0"}]},{"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: Solving Trigonometric Equations with Double Angles"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve equations involving double angles."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"cos 2x: Double angle formula for cosine"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To solve "},{"type":"inlineMath","attrs":{"latex":"a \\cos 2x + b = 0"}},{"type":"text","text":", isolate "},{"type":"inlineMath","attrs":{"latex":"\\cos 2x"}},{"type":"text","text":" 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":"Isolate "},{"type":"inlineMath","attrs":{"latex":"\\cos 2x"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"2 \\cos 2x - 1 = 0 \\implies 2 \\cos 2x = 1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide both sides by 2: "},{"type":"inlineMath","attrs":{"latex":"\\cos 2x = \\frac{1}{2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find all "},{"type":"inlineMath","attrs":{"latex":"2x"}},{"type":"text","text":" in "},{"type":"inlineMath","attrs":{"latex":"[0, 2\\pi]"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"\\cos 2x = \\frac{1}{2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For each solution for "},{"type":"inlineMath","attrs":{"latex":"2x"}},{"type":"text","text":", solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" in "},{"type":"inlineMath","attrs":{"latex":"[0, 2\\pi]"}},{"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 solutions are "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{\\pi}{6}"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{5\\pi}{6}"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{7\\pi}{6}"}},{"type":"text","text":", and "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{11\\pi}{6}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q9. Find all solutions of the inequality between 0 and 2π: 1 − 2 sin x > 0"}]},{"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: Solving Trigonometric Inequalities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve inequalities involving trigonometric functions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"sin x: Sine function"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To solve "},{"type":"inlineMath","attrs":{"latex":"a \\sin x + b > 0"}},{"type":"text","text":", isolate "},{"type":"inlineMath","attrs":{"latex":"\\sin x"}},{"type":"text","text":" and find the intervals where the inequality holds."}]}]}]},{"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":"Isolate "},{"type":"inlineMath","attrs":{"latex":"\\sin x"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"1 - 2 \\sin x > 0 \\implies -2 \\sin x > -1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide both sides by -2 (remember to reverse the inequality): "},{"type":"inlineMath","attrs":{"latex":"\\sin x < \\frac{1}{2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find all "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" in "},{"type":"inlineMath","attrs":{"latex":"[0, 2\\pi]"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"\\sin x < \\frac{1}{2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express the solution as intervals or a union of intervals."}]}]}]},{"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 solution is "},{"type":"inlineMath","attrs":{"latex":"x \\in (0, \\frac{\\pi}{6}) \\cup (\\frac{5\\pi}{6}, 2\\pi)"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"These are the intervals where "},{"type":"inlineMath","attrs":{"latex":"\\sin x < \\frac{1}{2}"}},{"type":"text","text":" in "},{"type":"inlineMath","attrs":{"latex":"[0, 2\\pi]"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q10. Find all solutions of the inequality between 0 and 2π: √3 cot x − 1 ≤ 0"}]},{"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: Solving Trigonometric Inequalities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve inequalities involving the cotangent function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"cot x: Cotangent function"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To solve "},{"type":"inlineMath","attrs":{"latex":"a \\cot x + b \\leq 0"}},{"type":"text","text":", isolate "},{"type":"inlineMath","attrs":{"latex":"\\cot x"}},{"type":"text","text":" and find the intervals where the inequality holds."}]}]}]},{"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":"Isolate "},{"type":"inlineMath","attrs":{"latex":"\\cot x"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{3} \\cot x - 1 \\leq 0 \\implies \\sqrt{3} \\cot x \\leq 1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide both sides by "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{3}"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"\\cot x \\leq \\frac{1}{\\sqrt{3}}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that "},{"type":"inlineMath","attrs":{"latex":"\\cot x = \\frac{1}{\\tan x}"}},{"type":"text","text":", so "},{"type":"inlineMath","attrs":{"latex":"\\tan x \\geq \\sqrt{3}"}},{"type":"text","text":" (since the inequality reverses for positive values)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find all "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" in "},{"type":"inlineMath","attrs":{"latex":"[0, 2\\pi]"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"\\tan x \\geq \\sqrt{3}"}},{"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 solution is "},{"type":"inlineMath","attrs":{"latex":"x \\in [\\frac{\\pi}{3}, \\frac{2\\pi}{3}] \\cup [\\frac{4\\pi}{3}, \\frac{5\\pi}{3}]"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q11. Evaluate without calculator, show all the supporting pictures: arccos(−√2/2)"}]},{"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: Inverse Trigonometric Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the arccosine function and your ability to find the angle whose cosine is a given value."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"arccos: Inverse cosine function, "},{"type":"inlineMath","attrs":{"latex":"\\arccos y"}},{"type":"text","text":" is the angle in "},{"type":"inlineMath","attrs":{"latex":"[0, \\pi]"}},{"type":"text","text":" whose cosine is "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Unit circle values for "},{"type":"inlineMath","attrs":{"latex":"\\cos 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":"Recall that "},{"type":"inlineMath","attrs":{"latex":"\\arccos y"}},{"type":"text","text":" gives the principal value in "},{"type":"inlineMath","attrs":{"latex":"[0, \\pi]"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the angle "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" in "},{"type":"inlineMath","attrs":{"latex":"[0, \\pi]"}},{"type":"text","text":" such that "},{"type":"inlineMath","attrs":{"latex":"\\cos x = -\\frac{\\sqrt{2}}{2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall the standard angles where "},{"type":"inlineMath","attrs":{"latex":"\\cos x = \\pm \\frac{\\sqrt{2}}{2}"}},{"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":"inlineMath","attrs":{"latex":"\\arccos(-\\frac{\\sqrt{2}}{2}) = \\frac{3\\pi}{4}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This is the principal value in "},{"type":"inlineMath","attrs":{"latex":"[0, \\pi]"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"\\cos x = -\\frac{\\sqrt{2}}{2}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q12. Evaluate without calculator: tan(tan⁻¹ 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: Inverse Trigonometric Functions and Function Composition"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the relationship between a function and its inverse."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\tan^{-1} y"}},{"type":"text","text":" is the angle whose tangent is "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\tan(\\tan^{-1} y) = y"}},{"type":"text","text":" for all real "},{"type":"inlineMath","attrs":{"latex":"y"}},{"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":"Let "},{"type":"inlineMath","attrs":{"latex":"\\theta = \\tan^{-1} 5"}},{"type":"text","text":", so "},{"type":"inlineMath","attrs":{"latex":"\\tan \\theta = 5"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Now, "},{"type":"inlineMath","attrs":{"latex":"\\tan(\\tan^{-1} 5) = \\tan(\\theta)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall the property of inverse functions: "},{"type":"inlineMath","attrs":{"latex":"f(f^{-1}(x)) = 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":"inlineMath","attrs":{"latex":"\\tan(\\tan^{-1} 5) = 5"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This is true for all real numbers "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q13. 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Evaluate without calculator: sec(arccot 2)"}]},{"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: Inverse Trigonometric Functions and Trigonometric Identities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use right triangle relationships to evaluate trigonometric expressions involving inverse functions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\arccot y"}},{"type":"text","text":" is the angle whose cotangent is "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sec \\theta = \\frac{1}{\\cos \\theta}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use a right triangle to relate cotangent and cosine."}]}]}]},{"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":"Let "},{"type":"inlineMath","attrs":{"latex":"\\theta = \\arccot 2"}},{"type":"text","text":", so "},{"type":"inlineMath","attrs":{"latex":"\\cot \\theta = 2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Draw a right triangle where the adjacent side is 2 and the opposite side is 1."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the hypotenuse using the Pythagorean theorem: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{2^2 + 1^2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find "},{"type":"inlineMath","attrs":{"latex":"\\cos \\theta"}},{"type":"text","text":" and then "},{"type":"inlineMath","attrs":{"latex":"\\sec \\theta"}},{"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":"inlineMath","attrs":{"latex":"\\sec(\\arccot 2) = \\frac{\\sqrt{5}}{2}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This comes from "},{"type":"inlineMath","attrs":{"latex":"\\cos \\theta = \\frac{2}{\\sqrt{5}}"}},{"type":"text","text":", so "},{"type":"inlineMath","attrs":{"latex":"\\sec \\theta = \\frac{\\sqrt{5}}{2}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q15. Evaluate without calculator: arctan(−1)"}]},{"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: Inverse Trigonometric Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the arctangent function and its range."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\arctan y"}},{"type":"text","text":" is the angle in "},{"type":"inlineMath","attrs":{"latex":"(-\\frac{\\pi}{2}, \\frac{\\pi}{2})"}},{"type":"text","text":" whose tangent is "},{"type":"inlineMath","attrs":{"latex":"y"}},{"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":"Find the angle "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" in "},{"type":"inlineMath","attrs":{"latex":"(-\\frac{\\pi}{2}, \\frac{\\pi}{2})"}},{"type":"text","text":" such that "},{"type":"inlineMath","attrs":{"latex":"\\tan x = -1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall the standard angles where "},{"type":"inlineMath","attrs":{"latex":"\\tan x = \\pm 1"}},{"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":"inlineMath","attrs":{"latex":"\\arctan(-1) = -\\frac{\\pi}{4}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This is the principal value in "},{"type":"inlineMath","attrs":{"latex":"(-\\frac{\\pi}{2}, \\frac{\\pi}{2})"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"\\tan x = -1"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q16. Evaluate without calculator: sin(sin⁻¹ 0.3)"}]},{"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: Inverse Trigonometric Functions and Function Composition"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the relationship between a function and its inverse."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin^{-1} y"}},{"type":"text","text":" is the angle in "},{"type":"inlineMath","attrs":{"latex":"[-\\frac{\\pi}{2}, \\frac{\\pi}{2}]"}},{"type":"text","text":" whose sine is "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin(\\sin^{-1} y) = y"}},{"type":"text","text":" for "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" in "},{"type":"inlineMath","attrs":{"latex":"[-1, 1]"}},{"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":"Let "},{"type":"inlineMath","attrs":{"latex":"\\theta = \\sin^{-1} 0.3"}},{"type":"text","text":", so "},{"type":"inlineMath","attrs":{"latex":"\\sin \\theta = 0.3"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Now, "},{"type":"inlineMath","attrs":{"latex":"\\sin(\\sin^{-1} 0.3) = \\sin(\\theta)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall the property of inverse functions: "},{"type":"inlineMath","attrs":{"latex":"f(f^{-1}(x)) = 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":"inlineMath","attrs":{"latex":"\\sin(\\sin^{-1} 0.3) = 0.3"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This is true for all "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" in "},{"type":"inlineMath","attrs":{"latex":"[-1, 1]"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q17. Evaluate without calculator: cos⁻¹(cos(7π/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: Inverse Trigonometric Functions and Function Composition"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the range of the inverse cosine function and how it interacts with the cosine function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos^{-1} y"}},{"type":"text","text":" has a range of "},{"type":"inlineMath","attrs":{"latex":"[0, \\pi]"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos^{-1}(\\cos x) = x"}},{"type":"text","text":" only if "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" is in "},{"type":"inlineMath","attrs":{"latex":"[0, \\pi]"}},{"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":"Note that "},{"type":"inlineMath","attrs":{"latex":"7\\pi/5"}},{"type":"text","text":" is not in "},{"type":"inlineMath","attrs":{"latex":"[0, \\pi]"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the equivalent angle in "},{"type":"inlineMath","attrs":{"latex":"[0, \\pi]"}},{"type":"text","text":" with the same cosine value as "},{"type":"inlineMath","attrs":{"latex":"7\\pi/5"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall the periodicity and symmetry of the cosine function."}]}]}]},{"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":"\\cos^{-1}(\\cos(\\frac{7\\pi}{5})) = \\frac{3\\pi}{5}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This is the equivalent angle in "},{"type":"inlineMath","attrs":{"latex":"[0, \\pi]"}},{"type":"text","text":" with the same cosine value."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q18. Evaluate without calculator: sin(arcsec 7)"}]},{"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: Inverse Trigonometric Functions and Trigonometric Identities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use right triangle relationships to evaluate trigonometric expressions involving inverse functions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\arcsec y"}},{"type":"text","text":" is the angle whose secant is "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin \\theta = \\frac{\\text{opposite}}{\\text{hypotenuse}}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use a right triangle to relate secant and sine."}]}]}]},{"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":"Let "},{"type":"inlineMath","attrs":{"latex":"\\theta = \\arcsec 7"}},{"type":"text","text":", so "},{"type":"inlineMath","attrs":{"latex":"\\sec \\theta = 7"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that "},{"type":"inlineMath","attrs":{"latex":"\\sec \\theta = \\frac{1}{\\cos \\theta}"}},{"type":"text","text":", so "},{"type":"inlineMath","attrs":{"latex":"\\cos \\theta = \\frac{1}{7}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Draw a right triangle with adjacent side 1 and hypotenuse 7."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the opposite side using the Pythagorean theorem: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{7^2 - 1^2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find "},{"type":"inlineMath","attrs":{"latex":"\\sin \\theta"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"\\frac{\\text{opposite}}{\\text{hypotenuse}}"}},{"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":"inlineMath","attrs":{"latex":"\\sin(\\arcsec 7) = \\frac{\\sqrt{48}}{7} = \\frac{4\\sqrt{3}}{7}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This comes from the triangle with sides 1, "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{48}"}},{"type":"text","text":", and 7."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q19. Solve the triangle if a = 1.23, A = 67°, B = 76°"}]},{"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: Solving Triangles (Law of Sines)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve a triangle given two angles and one side (AAS or ASA case)."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Law of Sines: "},{"type":"inlineMath","attrs":{"latex":"\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sum of angles in a triangle: "},{"type":"inlineMath","attrs":{"latex":"A + B + C = 180°"}}]}]}]},{"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":"Find angle "},{"type":"inlineMath","attrs":{"latex":"C"}},{"type":"text","text":" using "},{"type":"inlineMath","attrs":{"latex":"A + B + C = 180°"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the Law of Sines to set up "},{"type":"inlineMath","attrs":{"latex":"\\frac{a}{\\sin A} = \\frac{b}{\\sin B}"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the Law of Sines to set up "},{"type":"inlineMath","attrs":{"latex":"\\frac{a}{\\sin A} = \\frac{c}{\\sin C}"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Plug in the known values and set up the equations for "},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on y