Q1. Verify the identity: sec(−x) sin x = tan x

Background

Topic: Trigonometric Identities – Even-Odd, Reciprocal, and Quotient Identities

This question tests your ability to manipulate and verify trigonometric identities using fundamental properties and identities, such as even-odd identities, reciprocal identities, and quotient identities.

Key Terms and Formulas

• Even-Odd Identities: , ,

• Reciprocal Identity:

• Quotient Identity:

Step-by-Step Guidance

1. Start with the left side: .

2. Apply the even-odd identity for secant: , so the expression becomes .

3. Use the reciprocal identity: , so you have .

4. Combine the terms: .

Try solving on your own before revealing the answer!

Q2. Verify the identity:

Background

Topic: Trigonometric Identities – Reciprocal and Quotient Identities

This question asks you to verify a trigonometric identity by expressing all terms in terms of sine and cosine, and simplifying.

Key Terms and Formulas

• Reciprocal Identity:

• Quotient Identity:

Step-by-Step Guidance

1. Rewrite in terms of sine and cosine: .

2. Combine the terms over a common denominator: .

3. Consider what value of makes (or use a Pythagorean identity if needed).

Try solving on your own before revealing the answer!

Q3. Find the exact value of using sum and difference identities.

Background

Topic: Sum and Difference Formulas for Sine

This question tests your ability to use the sine sum or difference formula to find the exact value of a non-standard angle.

Key Terms and Formulas

• Sum and Difference Formula:

Step-by-Step Guidance

1. Express as a difference of two common angles: .

2. Apply the sine difference formula: .

3. Recall the exact values: , , , .

4. Substitute these values into the formula and simplify the expression, but do not combine into a single value yet.

Try solving on your own before revealing the answer!

Q4. Rewrite as a single trigonometric function and find its exact value.

Background

Topic: Sum and Difference Formulas for Sine

This question tests your ability to recognize and apply the sine sum formula to combine two terms into a single sine function.

Key Terms and Formulas

• Sine Sum Formula:

Step-by-Step Guidance

1. Identify and in the expression.

2. Apply the sine sum formula: .

3. Simplify the argument: .

4. Recall the exact value of .

Try solving on your own before revealing the answer!

Q5. Verify the identity:

Background

Topic: Trigonometric Angle Addition Formulas

This question tests your understanding of the cosine and sine addition formulas, and how to relate them for specific angle values.

Key Terms and Formulas

• Cosine Addition Formula:

• Values: ,

Step-by-Step Guidance

1. Rewrite using the cosine addition formula.

2. Substitute the known values for and .

3. Simplify the resulting expression to see if it matches .

Try solving on your own before revealing the answer!

Q6. Write as a sine, cosine, or tangent of a double angle and find its exact value.

Background

Topic: Double Angle Formulas

This question tests your ability to recognize and use the double angle formula for sine.

Key Terms and Formulas

• Sine Double Angle Formula:

Step-by-Step Guidance

1. Recognize that matches the double angle formula for sine.

2. Set and write .

3. Simplify to .

4. Recall the exact value for .

Try solving on your own before revealing the answer!

Q7. Write as a double angle and find its exact value.

Background

Topic: Double Angle Formulas for Cosine

This question tests your ability to use the double angle formula for cosine to rewrite and evaluate an expression.

Key Terms and Formulas

• Cosine Double Angle Formula:

Step-by-Step Guidance

1. Recognize that matches the double angle formula for cosine.

2. Set and write .

3. Simplify to .

4. Recall the exact value for .

Try solving on your own before revealing the answer!

Q8. Use a half-angle formula to find the exact value of .

Background

Topic: Half-Angle Formulas

This question tests your ability to use the half-angle formula for sine to find the exact value of a non-standard angle.

Key Terms and Formulas

• Half-Angle Formula for Sine:

Step-by-Step Guidance

1. Express as half of : .

2. Apply the half-angle formula: .

3. Determine the sign based on the quadrant (since is in the second quadrant, sine is positive).

4. Recall the exact value for and substitute it into the formula, but do not simplify further yet.

Try solving on your own before revealing the answer!

Q9. Use a half-angle formula to find the exact value of .

Background

Topic: Half-Angle Formulas

This question tests your ability to use the half-angle formula for cosine to find the exact value of a non-standard angle.

Key Terms and Formulas

• Half-Angle Formula for Cosine:

Step-by-Step Guidance

1. Express as half of : .

2. Apply the half-angle formula: .

3. Determine the sign based on the quadrant (since is in the first quadrant, cosine is positive).

4. Recall the exact value for and substitute it into the formula, but do not simplify further yet.

Try solving on your own before revealing the answer!

Q10. Express as a product and find its exact value.

Background

Topic: Sum-to-Product Formulas

This question tests your ability to use the sum-to-product identities to rewrite a sum of sines as a product, and then evaluate the result.

Key Terms and Formulas

• Sum-to-Product Formula:

Step-by-Step Guidance

1. Identify and .

2. Apply the sum-to-product formula: .

3. Simplify the arguments: , .

4. Recall the exact values for and and substitute, but do not multiply out yet.

Try solving on your own before revealing the answer!

Q11. Verify the identity:

Background

Topic: Sum-to-Product and Quotient Identities

This question tests your ability to use sum-to-product identities for sine and cosine, and then simplify the resulting quotient.

Key Terms and Formulas

• Sum-to-Product for Sine:

• Sum-to-Product for Cosine:

• Quotient Identity:

Step-by-Step Guidance

1. Apply the sum-to-product formula to the numerator: .

2. Apply the sum-to-product formula to the denominator: .

3. Substitute these expressions into the original fraction and simplify by canceling common factors.

4. Use the quotient identity to relate the result to .

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Q12. Solve for .

Background

Topic: Solving Trigonometric Equations

This question tests your ability to solve basic trigonometric equations for all solutions.

Key Terms and Formulas

• Basic algebraic manipulation

• Inverse sine function

Step-by-Step Guidance

1. Isolate by moving all terms involving to one side: .

2. Simplify to .

3. Divide both sides by $2\sin \theta$.

4. Find all solutions for in using the inverse sine function and the unit circle.

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Q13. Solve for in .

Background

Topic: Solving Trigonometric Equations with Double Angles

This question tests your ability to solve equations involving double angles and to find all solutions in a given interval.

Key Terms and Formulas

• Double angle:

• Inverse sine function

Step-by-Step Guidance

1. Set and find all solutions for in .

2. Recall that at , and add for periodicity.

3. Divide each solution for by $2\theta[0, 2\pi)$.

4. List all solutions for in the required interval.

Try solving on your own before revealing the answer!

Q14. Solve for in .

Background

Topic: Solving Trigonometric Equations

This question tests your ability to solve trigonometric equations involving both sine and cosine.

Key Terms and Formulas

• Algebraic manipulation

• Possible use of identities or substitution

Step-by-Step Guidance

1. Isolate one trigonometric function, for example, .

2. Divide both sides by $2\sin x = \frac{\cos x + 1}{2}$.

3. Consider possible values for that satisfy this equation, or use a substitution such as and if needed.

4. Solve for in , checking all possible solutions.

Try solving on your own before revealing the answer!

Q15. Solve for in .

Background

Topic: Solving Trigonometric Equations with Double Angles

This question tests your ability to use double angle identities and solve trigonometric equations.

Key Terms and Formulas

• Double Angle Formula:

Step-by-Step Guidance

1. Rewrite using the double angle formula: .

2. Move all terms to one side: .

3. Let and solve the quadratic equation for .

4. Find all in that satisfy (the roots you found).

Try solving on your own before revealing the answer!