Q1. Find the exact value of sin(105°) using the fact that 105° = 60° + 45°.

Background

Topic: Sum and Difference Formulas for Sine

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

Key Terms and Formulas

• Sum Formula for Sine:

• Reference angles: 60° and 45° are special angles with known sine and cosine values.

Step-by-Step Guidance

1. Express as .

2. Apply the sum formula: .

3. Recall the exact values: , , , .

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

Try solving on your own before revealing the answer!

Q2. Find the exact value of cos(67.5°) using the fact that .

Background

Topic: Half-Angle Formulas

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

Key Terms and Formulas

• Cosine Half-Angle Formula:

• Sign depends on the quadrant of .

Step-by-Step Guidance

1. Recognize that , so .

2. Use the half-angle formula: (determine the correct sign for the quadrant).

3. Recall .

4. Substitute this value into the formula and simplify, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q3. Find the exact value of tan(195°) using the fact that 195° = 45° + 150°.

Background

Topic: Sum and Difference Formulas for Tangent

This question tests your ability to use the tangent addition formula to find the exact value of a non-standard angle.

Key Terms and Formulas

• Tangent Sum Formula:

• Reference angles: ,

Step-by-Step Guidance

1. Express as .

2. Apply the tangent sum formula: .

3. Substitute the known values for and .

4. Simplify the numerator and denominator, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q4. Find the exact value of using the fact that .

Background

Topic: Sum and Difference Formulas for Cosine (Radians)

This question tests your ability to use the cosine addition formula with radian measures.

Key Terms and Formulas

• Cosine Sum Formula:

• Reference angles: and

Step-by-Step Guidance

1. Express as .

2. Apply the sum formula: .

3. Recall the exact values for and of and .

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

Try solving on your own before revealing the answer!

Q5. Find the exact value of using the fact that .

Background

Topic: Negative Angles and Half-Angle Formulas

This question tests your understanding of sine for negative angles and possibly the half-angle formula.

Key Terms and Formulas

• Odd Function Property:

• Half-Angle Formula:

Step-by-Step Guidance

1. Recognize that is a negative angle; use the odd property of sine if helpful.

2. Express in terms of a known angle, such as or use the half-angle formula if appropriate.

3. Recall the exact value for (or , since cosine is even).

4. Set up the half-angle formula and substitute the known value, but do not compute the final value yet.

Try solving on your own before revealing the answer!