Precalculus Trigonometry Review – Step-by-Step Guidance

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

Background

Topic: Trigonometric Angle Addition Formulas

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

Key Terms and Formulas

• Angle Addition Formula for Sine: $\sin(A+B) = \sin A \cos B + \cos A \sin B$

• Reference values: $\sin 60^\circ, \cos 60^\circ, \sin 45^\circ, \cos 45^\circ$

Step-by-Step Guidance

1. Identify $A = 60^\circ$ and $B = 45^\circ$.

2. Write the angle addition formula for sine: $\sin(105^\circ) = \sin(60^\circ + 45^\circ) = \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ$.

3. Recall the exact values: $\sin 60^\circ = \frac{\sqrt{3}}{2}$, $\cos 60^\circ = \frac{1}{2}$, $\sin 45^\circ = \frac{\sqrt{2}}{2}$, $\cos 45^\circ = \frac{\sqrt{2}}{2}$.

4. Substitute these values into the formula, but do not simplify yet.

Try solving on your own before revealing the answer!

Final Answer: $\frac{\sqrt{6} + \sqrt{2}}{4}$

After substituting and simplifying, you get $\sin(105^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4}$.

Q2. Find the exact value of cos(67.5°) using the fact that $67.5^\circ = \frac{135^\circ}{2}$.

Background

Topic: Half-Angle Formulas

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

Key Terms and Formulas

• Cosine Half-Angle Formula: $\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}$

• Reference value: $\cos 135^\circ$

Step-by-Step Guidance

1. Recognize that $67.5^\circ = \frac{135^\circ}{2}$, so $\theta = 135^\circ$.

2. Write the half-angle formula: $\cos(67.5^\circ) = \cos\left(\frac{135^\circ}{2}\right) = \pm \sqrt{\frac{1 + \cos 135^\circ}{2}}$.

3. Determine the sign: Since $67.5^\circ$ is in the first quadrant, cosine is positive.

4. Recall $\cos 135^\circ = -\frac{\sqrt{2}}{2}$ and substitute into the formula, but do not simplify yet.

Try solving on your own before revealing the answer!

Final Answer: $\frac{\sqrt{2 + \sqrt{2}}}{2}$

After substituting and simplifying, $\cos(67.5^\circ) = \frac{\sqrt{2 + \sqrt{2}}}{2}$.

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

Background

Topic: Trigonometric Angle Addition Formulas

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

Key Terms and Formulas

• Tangent Addition Formula: $\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

• Reference values: $\tan 45^\circ, \tan 150^\circ$

Step-by-Step Guidance

1. Identify $A = 45^\circ$ and $B = 150^\circ$.

2. Write the tangent addition formula: $\tan(195^\circ) = \tan(45^\circ + 150^\circ) = \frac{\tan 45^\circ + \tan 150^\circ}{1 - \tan 45^\circ \tan 150^\circ}$.

3. Recall $\tan 45^\circ = 1$ and $\tan 150^\circ = -\frac{1}{\sqrt{3}}$.

4. Substitute these values into the formula, but do not simplify yet.

Try solving on your own before revealing the answer!

Final Answer: $-2 + \sqrt{3}$

After substituting and simplifying, $\tan(195^\circ) = -2 + \sqrt{3}$.

Q4. Find the exact value of $\cos\left(\frac{11\pi}{12}\right)$ using the fact that $\frac{11\pi}{12} = \frac{2\pi}{3} + \frac{\pi}{4}$.

Background

Topic: Trigonometric Angle Addition Formulas (Radians)

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

Key Terms and Formulas

• Cosine Addition Formula: $\cos(A+B) = \cos A \cos B - \sin A \sin B$

• Reference values: $\cos \frac{2\pi}{3}, \sin \frac{2\pi}{3}, \cos \frac{\pi}{4}, \sin \frac{\pi}{4}$

Step-by-Step Guidance

1. Identify $A = \frac{2\pi}{3}$ and $B = \frac{\pi}{4}$.

2. Write the cosine addition formula: $\cos\left(\frac{11\pi}{12}\right) = \cos\left(\frac{2\pi}{3} + \frac{\pi}{4}\right) = \cos \frac{2\pi}{3} \cos \frac{\pi}{4} - \sin \frac{2\pi}{3} \sin \frac{\pi}{4}$.

3. Recall the exact values for these angles.

4. Substitute these values into the formula, but do not simplify yet.

Try solving on your own before revealing the answer!

Final Answer: $-\frac{\sqrt{6} + \sqrt{2}}{4}$

After substituting and simplifying, $\cos\left(\frac{11\pi}{12}\right) = -\frac{\sqrt{6} + \sqrt{2}}{4}$.

Q5. Find the exact value of $\sin\left(-\frac{\pi}{12}\right)$ using the fact that $-\frac{\pi}{12} = -\frac{\pi}{6} \times \frac{1}{2}$.

Background

Topic: Trigonometric Angle Multiplication and Half-Angle Formulas

This question tests your ability to use the sine half-angle formula for negative angles.

Key Terms and Formulas

• Sine Half-Angle Formula: $\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}$

• Reference value: $\cos(-\frac{\pi}{6})$

Step-by-Step Guidance

1. Recognize that $-\frac{\pi}{12} = \frac{1}{2} \times \left(-\frac{\pi}{6}\right)$, so $\theta = -\frac{\pi}{6}$.

2. Write the half-angle formula: $\sin\left(-\frac{\pi}{12}\right) = \sin\left(\frac{1}{2} \times -\frac{\pi}{6}\right) = \pm \sqrt{\frac{1 - \cos(-\frac{\pi}{6})}{2}}$.

3. Determine the sign: Since $-\frac{\pi}{12}$ is negative, sine is negative.

4. Recall $\cos(-\frac{\pi}{6}) = \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$ and substitute into the formula, but do not simplify yet.

Try solving on your own before revealing the answer!

Final Answer: $-\frac{\sqrt{2 - \sqrt{3}}}{2}$

After substituting and simplifying, $\sin\left(-\frac{\pi}{12}\right) = -\frac{\sqrt{2 - \sqrt{3}}}{2}$.