Q1. Identify the amplitude, frequency, period, and shifts of the equation

Background

Topic: Trigonometric Function Transformations

This question tests your understanding of how the parameters in a cosine function affect its graph, including amplitude, period, frequency, and horizontal/vertical shifts.

Key Terms and Formulas:

• Amplitude: in

• Period:

• Frequency:

• Horizontal shift:

• Vertical shift:

Step-by-Step Guidance

1. Compare the given equation to the standard form .

2. Identify , , , and from the equation: .

3. Determine the amplitude by taking the absolute value of .

4. Calculate the period using .

5. Find the frequency as .

6. Set up the formulas for horizontal and vertical shifts, but stop before plugging in the values.

Try solving on your own before revealing the answer!

Q2. Graph one basic period of the equation

Background

Topic: Graphing Trigonometric Functions

This question asks you to sketch the graph of a transformed cosine function over one basic period, using the amplitude, period, and shifts identified previously.

Key Terms and Formulas:

• Basic period: or adjusted for horizontal shift

• Key points: maximum, minimum, midline

• Vertical shift: moves the midline up/down

Step-by-Step Guidance

1. Recall the amplitude, period, and shifts from Q1.

2. Determine the starting and ending -values for one basic period, considering the horizontal shift.

3. Identify the midline of the graph (vertical shift).

4. Mark the maximum and minimum points based on amplitude and vertical shift.

5. Set up the -values for key points (start, quarter period, half period, etc.), but do not plot the final graph yet.

Try solving on your own before revealing the answer!

Q3. Graph one basic period of the equation

Background

Topic: Graphing Secant Functions

This question tests your ability to graph the secant function, which is the reciprocal of cosine, using the same transformations as in Q2.

Key Terms and Formulas:

• Vertical asymptotes occur where

• Use the cosine graph as a guide for secant

Step-by-Step Guidance

1. Identify the period, amplitude, and shifts as in Q1 and Q2.

2. Graph the corresponding cosine function as a reference.

3. Mark vertical asymptotes where the cosine function crosses zero.

4. Sketch the secant curves (U-shaped branches) above and below the midline, but do not complete the graph yet.

Try solving on your own before revealing the answer!

Q4. Identify the amplitude, frequency, period, and shifts of the equation

Background

Topic: Trigonometric Function Transformations

This question tests your understanding of how the parameters in a sine function affect its graph, including amplitude, period, frequency, and horizontal/vertical shifts.

Key Terms and Formulas:

• Amplitude: in

• Period:

• Frequency:

• Horizontal shift:

• Vertical shift:

Step-by-Step Guidance

1. Compare the equation to the standard form .

2. Identify , , , and from the equation.

3. Set up the formulas for amplitude, period, frequency, and shifts.

4. Prepare to plug in the values, but stop before calculating the final numbers.

Try solving on your own before revealing the answer!

Q5. Graph one basic period of the equation

Background

Topic: Graphing Sine Functions

This question asks you to sketch the graph of a transformed sine function over one basic period, using the amplitude, period, and shifts identified previously.

Key Terms and Formulas:

• Basic period: or adjusted for horizontal shift

• Key points: maximum, minimum, midline

• Vertical shift: moves the midline up/down

Step-by-Step Guidance

1. Recall the amplitude, period, and shifts from Q4.

2. Determine the starting and ending -values for one basic period, considering the horizontal shift.

3. Identify the midline of the graph (vertical shift).

4. Mark the maximum and minimum points based on amplitude and vertical shift.

5. Set up the -values for key points (start, quarter period, half period, etc.), but do not plot the final graph yet.

Try solving on your own before revealing the answer!

Q6. Graph one basic period of the equation

Background

Topic: Graphing Cosecant Functions

This question tests your ability to graph the cosecant function, which is the reciprocal of sine, using the same transformations as in Q5.

Key Terms and Formulas:

• Vertical asymptotes occur where

• Use the sine graph as a guide for cosecant

Step-by-Step Guidance

1. Identify the period, amplitude, and shifts as in Q4 and Q5.

2. Graph the corresponding sine function as a reference.

3. Mark vertical asymptotes where the sine function crosses zero.

4. Sketch the cosecant curves (U-shaped branches) above and below the midline, but do not complete the graph yet.

Try solving on your own before revealing the answer!

Q7. Without calculator, express cot as a function of its reference angle and find the exact value

Background

Topic: Reference Angles and Exact Trigonometric Values

This question tests your ability to use reference angles and the unit circle to find exact values of trigonometric functions.

Key Terms and Formulas:

• Reference angle: The acute angle formed with the x-axis

• Cotangent:

• Quadrant signs: cotangent is positive in Quadrant IV

Step-by-Step Guidance

1. Find the reference angle for .

2. Determine the sign of cotangent in the quadrant where lies.

3. Express in terms of the reference angle.

4. Set up the exact value using known values from the unit circle, but stop before calculating the final result.

Try solving on your own before revealing the answer!

Q8. Without calculator, express as a function of its reference angle and find the exact value

Background

Topic: Reference Angles and Exact Trigonometric Values

This question tests your ability to use reference angles and the unit circle to find exact values of sine.

Key Terms and Formulas:

• Reference angle: for angles in Quadrant II

• Sine: positive in Quadrant II

• Exact values:

Step-by-Step Guidance

1. Find the reference angle for .

2. Determine the sign of sine in the quadrant where lies.

3. Express in terms of the reference angle.

4. Set up the exact value using known values from the unit circle, but stop before calculating the final result.

Try solving on your own before revealing the answer!

Q9. The terminal side of an angle is given by , . Determine the values of all six trigonometric functions of the angle

Background

Topic: Trigonometric Functions from Coordinates

This question tests your ability to find sine, cosine, tangent, and their reciprocals given a line and a restriction on .

Key Terms and Formulas:

• Line equation:

• Choose a point on the terminal side with (e.g., )

• Trigonometric functions: , , ,

Step-by-Step Guidance

1. Pick a point on the terminal side, such as , .

2. Calculate .

3. Set up the formulas for , , , , , .

4. Prepare to plug in the values, but stop before calculating the final numbers.

Try solving on your own before revealing the answer!

Q10. Verify the identity

Background

Topic: Trigonometric Identities

This question tests your ability to manipulate and verify trigonometric identities using algebraic and trigonometric properties.

Key Terms and Formulas:

• Basic identities:

• Factoring and combining like terms

Step-by-Step Guidance

1. Start with the left side: .

2. Factor from both terms.

3. Express in terms of using the Pythagorean identity.

4. Set up the simplification, but stop before completing the identity.

Try solving on your own before revealing the answer!

Q11. Verify the identity

Background

Topic: Trigonometric Identities

This question tests your ability to manipulate and verify trigonometric identities using algebraic and trigonometric properties.

Key Terms and Formulas:

Step-by-Step Guidance

1. Start with the left side: .

2. Expand the expression using the definitions of secant and tangent.

3. Write each term with a common denominator.

4. Set up the simplification, but stop before completing the identity.

Try solving on your own before revealing the answer!

Q12. Find if and

Background

Topic: Trigonometric Functions in Quadrants

This question tests your ability to find the value of cosine given sine and the quadrant.

Key Terms and Formulas:

• Pythagorean identity:

• Quadrant IV: cosine is positive

Step-by-Step Guidance

1. Plug into the Pythagorean identity.

2. Solve for .

3. Take the square root, considering the sign of cosine in Quadrant IV.

4. Set up the expression for , but stop before calculating the final value.

Try solving on your own before revealing the answer!

Q13. Find the exact value without calculator,

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Topic: Exact Trigonometric Values and Sum/Difference Formulas

This question tests your ability to use sum or difference formulas to find exact values for non-standard angles.

Key Terms and Formulas:

• Cosine sum formula:

• Use

• Exact values for , , ,

Step-by-Step Guidance

1. Express as .

2. Write out the sum formula for cosine.

3. Plug in the exact values for , , , .

4. Set up the calculation, but stop before simplifying to the final value.

Try solving on your own before revealing the answer!

Q14. Find the exact value without calculator,

Background

Topic: Exact Trigonometric Values and Sum/Difference Formulas

This question tests your ability to use sum or difference formulas to find exact values for non-standard angles.

Key Terms and Formulas:

• Sine difference formula:

• Use

• Exact values for , , ,

Step-by-Step Guidance

1. Express as .

2. Write out the difference formula for sine.

3. Plug in the exact values for , , , .

4. Set up the calculation, but stop before simplifying to the final value.

Try solving on your own before revealing the answer!

Q15. Solve the right triangle if , (Round your answer to the appropriate value!)

Background

Topic: Solving Right Triangles

This question tests your ability to use trigonometric ratios to solve for unknown sides and angles in a right triangle.

Key Terms and Formulas:

• Let be one acute angle, the hypotenuse

• Use trigonometric ratios to solve for other sides

Step-by-Step Guidance

1. Write and , where and are the legs.

2. Set up the equations to solve for and .

3. Find the other angle using .

4. Prepare to plug in the values, but stop before calculating the final numbers.

Try solving on your own before revealing the answer!

Q16. Solve the right triangle if , (Round your answer to the appropriate value!)

Background

Topic: Solving Right Triangles

This question tests your ability to use trigonometric ratios to solve for unknown sides and angles in a right triangle.

Key Terms and Formulas:

• Use inverse trigonometric functions to find angles

Step-by-Step Guidance

1. Write to solve for angle .

2. Use .

3. Find the other angle using .

4. Set up , but stop before calculating the final numbers.

Try solving on your own before revealing the answer!