Precalculus Practice Test 3 (Trigonometric Equations, Identities, and Formulas) – Step-by-Step Guidance

Q1. Find all solutions to $\sin \theta = \frac{1}{2}$ and give the general formula and specific solutions on $[0, 2\pi)$.

Background

Topic: Solving Trigonometric Equations

This question tests your ability to solve basic trigonometric equations, find all solutions using the unit circle, and express the general solution in radians.

Key Terms and Formulas:

• $\sin \theta = \frac{1}{2}$

• General solution for $\sin \theta = a$: $\theta = \arcsin(a) + 2\pi k$ or $\theta = \pi - \arcsin(a) + 2\pi k$, where $k$ is any integer.

Step-by-Step Guidance

1. Recall the values of $\theta$ in $[0, 2\pi)$ where $\sin \theta = \frac{1}{2}$. Use the unit circle or known reference angles.

2. Write the general solution using $\arcsin\left(\frac{1}{2}\right)$ and $\pi - \arcsin\left(\frac{1}{2}\right)$, adding $2\pi k$ to each.

3. Identify the specific solutions in $[0, 2\pi)$ by evaluating the reference angles and checking which quadrants sine is positive.

4. Express your answers in radians, using the smallest non-negative angles.

Try solving on your own before revealing the answer!

General Solution: $\theta = \frac{\pi}{6} + 2\pi k$ or $\theta = \frac{5\pi}{6} + 2\pi k$

Specific solutions on $[0, 2\pi)$: $\theta = \frac{\pi}{6}, \frac{5\pi}{6}$

These are the two angles in $[0, 2\pi)$ where the sine value is $\frac{1}{2}$.

Q2. Verify the identity: $\frac{1 + \sin t}{\cos t} = \frac{\cos t}{1 - \sin t}$

Background

Topic: Verifying Trigonometric Identities

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

Key Terms and Formulas:

• Pythagorean Identity: $\sin^2 t + \cos^2 t = 1$

• Algebraic manipulation: multiplying numerator and denominator by a conjugate

Step-by-Step Guidance

1. Start with the left side: $\frac{1 + \sin t}{\cos t}$.

2. Multiply numerator and denominator by the conjugate of the denominator, $1 - \sin t$, to rationalize.

3. Simplify the numerator: $(1 + \sin t)(1 - \sin t) = 1 - \sin^2 t$.

4. Recall that $1 - \sin^2 t = \cos^2 t$ and substitute.

Try solving on your own before revealing the answer!

Final Identity: $\frac{\cos t}{1 - \sin t}$

After simplifying, both sides are equal, confirming the identity.

Q3. Rewrite $\frac{2 \tan \frac{5\pi}{8}}{1 - \tan^2 \frac{5\pi}{8}}$ as a double-angle expression and find its exact value.

Background

Topic: Double-Angle Formulas for Tangent

This question tests your ability to recognize and use the double-angle formula for tangent to simplify and evaluate trigonometric expressions.

Key Terms and Formulas:

• Double-angle formula for tangent: $\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}$

Step-by-Step Guidance

1. Recognize that the given expression matches the double-angle formula for tangent.

2. Let $x = \frac{5\pi}{8}$, so $2x = \frac{5\pi}{4}$.

3. Rewrite the expression as $\tan \left(2 \cdot \frac{5\pi}{8}\right) = \tan \frac{5\pi}{4}$.

4. Find the exact value of $\tan \frac{5\pi}{4}$ using the unit circle.

Try solving on your own before revealing the answer!

Double-Angle Expression: $\tan \frac{5\pi}{4}$

Exact value: $1$

This is because $\tan \frac{5\pi}{4}$ is in the third quadrant, where tangent is positive and equals $1$.

Q4. Verify the identity: $\frac{\tan x + 1}{\sec x} = \cos x + \sin x$

Background

Topic: Verifying Trigonometric Identities

This question tests your ability to manipulate trigonometric expressions and use reciprocal and quotient identities.

Key Terms and Formulas:

• $\tan x = \frac{\sin x}{\cos x}$

• $\sec x = \frac{1}{\cos x}$

Step-by-Step Guidance

1. Rewrite $\tan x$ and $\sec x$ in terms of sine and cosine.

2. Combine the numerator: $\tan x + 1 = \frac{\sin x}{\cos x} + 1 = \frac{\sin x + \cos x}{\cos x}$.

3. Divide by $\sec x$: $\frac{\sin x + \cos x}{\cos x} \div \frac{1}{\cos x}$.

4. Simplify the division to get $\sin x + \cos x$.

Try solving on your own before revealing the answer!

Final Identity: $\cos x + \sin x$

Both sides are equal after simplification, confirming the identity.

Q5. Verify the identity: $\frac{\sec 2x - \tan 2x}{\sec x + \tan x} = \sec x - \tan x$

Background

Topic: Verifying Trigonometric Identities (Double-Angle)

This question tests your ability to use double-angle identities and algebraic manipulation to verify trigonometric identities.

Key Terms and Formulas:

• $\sec 2x = \frac{1}{\cos 2x}$

• $\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}$

• Difference of squares: $(a + b)(a - b) = a^2 - b^2$

Step-by-Step Guidance

1. Multiply numerator and denominator by $\sec x - \tan x$ to rationalize the denominator.

2. Use the identity $(\sec x + \tan x)(\sec x - \tan x) = \sec^2 x - \tan^2 x$.

3. Simplify the numerator using double-angle identities.

4. Show that the expression reduces to $\sec x - \tan x$.

Try solving on your own before revealing the answer!

Final Identity: $\sec x - \tan x$

After simplification, both sides are equal, confirming the identity.

Q6. Rewrite $\cos^2 \frac{\pi}{8} - \sin^2 \frac{\pi}{8}$ as a double-angle expression 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 trigonometric expressions.

Key Terms and Formulas:

• Double-angle formula: $\cos 2x = \cos^2 x - \sin^2 x$

Step-by-Step Guidance

1. Recognize that $\cos^2 \frac{\pi}{8} - \sin^2 \frac{\pi}{8}$ matches the double-angle formula for cosine.

2. Let $x = \frac{\pi}{8}$, so $2x = \frac{\pi}{4}$.

3. Rewrite the expression as $\cos \frac{\pi}{4}$.

4. Recall the exact value of $\cos \frac{\pi}{4}$.

Try solving on your own before revealing the answer!

Double-Angle Expression: $\cos \frac{\pi}{4}$

Exact value: $\frac{1}{\sqrt{2}}$ or $\frac{\sqrt{2}}{2}$