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

Background

Topic: Graphs of Trigonometric Functions

This question tests your understanding of how to analyze the properties of a transformed cosine function, including amplitude, period, frequency, and phase/vertical shifts.

Key Terms and Formulas:

• Amplitude: in

• Period:

• Frequency:

• Phase Shift:

• Vertical Shift:

Step-by-Step Guidance

1. Rewrite the equation in the standard form: .

2. Identify the values of , , , and from the equation.

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

4. Calculate the period using .

5. Determine the phase shift using and the vertical shift .

Try solving on your own before revealing the answer!

Q2. Graph one basic period of the equation

Background

Topic: Graphing Cosine Functions with Transformations

This question asks you to graph a transformed cosine function, showing one full period and all relevant shifts.

Key Terms and Formulas:

• Basic cosine graph:

• Amplitude, period, phase shift, and vertical shift as above

• Key points: maximum, minimum, and midline crossings

Step-by-Step Guidance

1. Determine the amplitude, period, phase shift, and vertical shift (from Q1).

2. Find the starting and ending -values for one period, considering the phase shift.

3. Calculate the -values at key points (maximum, minimum, midline) within one period.

4. Sketch the midline at and plot the transformed cosine curve accordingly.

5. Label the -intercepts, maximum, and minimum points for clarity.

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, including all transformations.

Key Terms and Formulas:

• Vertical asymptotes where

• Use the corresponding cosine graph as a guide

Step-by-Step Guidance

1. Graph the related cosine function as a reference.

2. Identify where the cosine function crosses the midline (these become the vertices of the secant graph).

3. Mark vertical asymptotes at -values where the cosine function equals zero.

4. Sketch the secant curves (U-shaped and inverted U-shaped) between the asymptotes, matching the amplitude and vertical shift.

5. Label the period, phase shift, and vertical shift on your graph.

Try solving on your own before revealing the answer!

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

Background

Topic: Graphs of Sine Functions with Transformations

This question asks you to analyze a transformed sine function for its key properties.

Key Terms and Formulas:

• Amplitude:

• Period:

• Frequency:

• Phase Shift:

• Vertical Shift:

Step-by-Step Guidance

1. Write the equation in the form .

2. Identify , , , and from the equation.

3. Find the amplitude, period, frequency, phase shift, and vertical shift using the formulas above.

4. Calculate the phase shift by evaluating .

5. Summarize all properties in a list.

Try solving on your own before revealing the answer!

Q5. Graph one basic period of the equation

Background

Topic: Graphing Sine Functions with Transformations

This question asks you to graph a sine function with amplitude, period, phase shift, and vertical shift.

Key Terms and Formulas:

• Amplitude, period, phase shift, and vertical shift as above

• Key points: maximum, minimum, and midline crossings

Step-by-Step Guidance

1. Determine the amplitude, period, phase shift, and vertical shift (from Q4).

2. Find the starting and ending -values for one period, considering the phase shift.

3. Calculate the -values at key points (maximum, minimum, midline) within one period.

4. Sketch the midline at and plot the transformed sine curve accordingly.

5. Label the -intercepts, maximum, and minimum points for clarity.

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, including all transformations.

Key Terms and Formulas:

• Vertical asymptotes where

• Use the corresponding sine graph as a guide

Step-by-Step Guidance

1. Graph the related sine function as a reference.

2. Identify where the sine function crosses the midline (these become the vertices of the cosecant graph).

3. Mark vertical asymptotes at -values where the sine function equals zero.

4. Sketch the cosecant curves (U-shaped and inverted U-shaped) between the asymptotes, matching the amplitude and vertical shift.

5. Label the period, phase shift, and vertical shift on your graph.

Try solving on your own before revealing the answer!

Q7. Without a 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 trigonometric functions.

Key Terms and Formulas:

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

• or

• Signs of trigonometric functions in different quadrants

Step-by-Step Guidance

1. Find the reference angle for .

2. Determine the sign of based on its quadrant.

3. Express in terms of the reference angle.

4. Recall the exact values for and .

5. Set up the expression for using these values, but do not compute the final value yet.

Try solving on your own before revealing the answer!

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

Background

Topic: Reference Angles and Exact Trigonometric Values (Radians)

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

Key Terms and Formulas:

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

• Signs of sine in different quadrants

• Exact values for

Step-by-Step Guidance

1. Find the reference angle for .

2. Determine the sign of based on its quadrant.

3. Express in terms of the reference angle.

4. Recall the exact value for .

5. Set up the expression for using these values, but do not compute the final value yet.

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 a Line Equation

This question tests your ability to find trigonometric function values given a line in the coordinate plane and a restriction on .

Key Terms and Formulas:

• Given a point on the terminal side,

• , ,

• Cosecant, secant, cotangent are reciprocals

Step-by-Step Guidance

1. Rewrite as .

2. Choose a point on this line with (e.g., , ).

3. Calculate for your chosen point.

4. Set up the expressions for , , , , , and using , , and .

5. Do not compute the final values yet; leave them in terms of , , and .

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 Pythagorean identities.

Key Terms and Formulas:

• Pythagorean identity:

• Factoring and simplifying expressions

Step-by-Step Guidance

1. Factor from both terms on the left side.

2. Express in terms of using the Pythagorean identity.

3. Simplify the expression inside the parentheses.

4. Combine like terms to see if the left side equals the right side.

5. Stop before the final simplification step.

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 definitions and algebraic manipulation.

Key Terms and Formulas:

Step-by-Step Guidance

1. Rewrite and in terms of sine and cosine.

2. Expand the left side: .

3. Simplify the expression to a single fraction.

4. Compare the result to the right side written as a single fraction.

5. Stop before the final simplification step.

Try solving on your own before revealing the answer!

Q12. Find if and

Background

Topic: Trigonometric Values in Different Quadrants

This question tests your ability to find one trigonometric function given another, using the Pythagorean identity and quadrant information.

Key Terms and Formulas:

• Pythagorean identity:

• Signs of cosine in the fourth quadrant

Step-by-Step Guidance

1. Plug into the Pythagorean identity.

2. Solve for .

3. Take the square root to find .

4. Determine the correct sign for based on the quadrant.

5. Stop before calculating the final value.

Try solving on your own before revealing the answer!

Q13. Find the exact value without a calculator:

Background

Topic: Exact Trigonometric Values Using Sum and Difference Formulas

This question tests your ability to use angle sum or difference identities to find exact values.

Key Terms and Formulas:

• Angle sum identity:

• Common angles: , ,

Step-by-Step Guidance

1. Express as a sum of two common angles (e.g., ).

2. Write out the angle sum formula for cosine.

3. Substitute the exact values for , , , and .

4. Set up the expression but do not simplify to a single value yet.

Try solving on your own before revealing the answer!

Q14. Find the exact value without a calculator:

Background

Topic: Exact Trigonometric Values Using Sum and Difference Formulas

This question tests your ability to use angle sum or difference identities to find exact values.

Key Terms and Formulas:

• Angle difference identity:

• Common angles: ,

Step-by-Step Guidance

1. Express as .

2. Write out the angle difference formula for sine.

3. Substitute the exact values for , , , and .

4. Set up the expression but do not simplify to a single value yet.

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.

• (opposite side)

• (adjacent side)

• Other angle

Step-by-Step Guidance

1. Calculate .

2. Find .

3. Find .

4. Set up the expressions for and but do not compute the final values yet.

Try solving on your own before revealing the answer!

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

Background

Topic: Solving Right Triangles

This question tests your ability to use trigonometric ratios and the Pythagorean theorem to solve for unknown sides and angles.

Key Terms and Formulas:

• is the side opposite angle , is the hypotenuse.

Step-by-Step Guidance

1. Calculate .

2. Find .

3. Calculate .

4. Set up the expressions for , , and but do not compute the final values yet.

Try solving on your own before revealing the answer!