Precalculus Chapter 6 Practice Test – Step-by-Step Guidance

Q1. Find the exact value of cos(112.5°)

Background

Topic: Trigonometric Values and Angle Sum/Difference Formulas

This question tests your ability to use trigonometric identities, specifically the half-angle or sum/difference formulas, to find the exact value of a cosine function for a non-standard angle.

Key Terms and Formulas:

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

• Sum/Difference Formula: $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

Step-by-Step Guidance

1. Recognize that $112.5^\circ$ is half of $225^\circ$, so you can use the half-angle formula.

2. Set $\theta = 225^\circ$, so $\cos(112.5^\circ) = \cos\left(\frac{225^\circ}{2}\right)$.

3. Apply the half-angle formula: $\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos\theta}{2}}$.

4. Determine the sign based on the quadrant: $112.5^\circ$ is in the second quadrant, where cosine is negative.

5. Find $\cos(225^\circ)$ using the unit circle or reference angles.

Try solving on your own before revealing the answer!

Q2. Find the exact value of $\sin\left(\frac{13\pi}{12}\right)$

Background

Topic: Trigonometric Values for Non-Standard Angles (Sum/Difference Formulas)

This question tests your ability to use the sum or difference formulas for sine to find the exact value of an angle not found on the unit circle.

Key Terms and Formulas:

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

• Difference Formula for Sine: $\sin(A - B) = \sin A \cos B - \cos A \sin B$

Step-by-Step Guidance

1. Express $\frac{13\pi}{12}$ as a sum or difference of angles with known sine and cosine values (e.g., $\frac{13\pi}{12} = \pi - \frac{\pi}{12}$).

2. Apply the sine difference formula: $\sin(\pi - x) = \sin\pi \cos x - \cos\pi \sin x$.

3. Recall the values: $\sin\pi = 0$, $\cos\pi = -1$.

4. Substitute $x = \frac{\pi}{12}$ and simplify the expression.

Try solving on your own before revealing the answer!

Q3. Find the exact value of $\tan 15^\circ$

Background

Topic: Tangent of Non-Standard Angles (Sum/Difference Formulas)

This question tests your ability to use the tangent difference formula to find the exact value of $\tan 15^\circ$.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Express $15^\circ$ as $45^\circ - 30^\circ$.

2. Apply the tangent difference formula: $\tan(45^\circ - 30^\circ) = \frac{\tan 45^\circ - \tan 30^\circ}{1 + \tan 45^\circ \tan 30^\circ}$.

3. Recall the values: $\tan 45^\circ = 1$, $\tan 30^\circ = \frac{1}{\sqrt{3}}$.

4. Substitute these values into the formula and simplify the numerator and denominator separately.

Try solving on your own before revealing the answer!

Q4. Prove the identity: $\tan\theta \cot\theta - \sin^2\theta = \cos^2\theta$

Background

Topic: Trigonometric Identities and Proofs

This question tests your ability to manipulate and prove trigonometric identities using fundamental relationships between the functions.

Key Terms and Formulas:

• $\tan\theta = \frac{\sin\theta}{\cos\theta}$

• $\cot\theta = \frac{\cos\theta}{\sin\theta}$

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

Step-by-Step Guidance

1. Rewrite $\tan\theta \cot\theta$ in terms of sine and cosine.

2. Simplify $\tan\theta \cot\theta$ to a single value.

3. Subtract $\sin^2\theta$ from your result.

4. Use the Pythagorean identity to relate $\sin^2\theta$ and $\cos^2\theta$.

Try solving on your own before revealing the answer!

Q5. Prove the identity: $\frac{\sec(x) - \tan(x)}{\sin(x)} = \frac{1}{\sec(x)}$

Background

Topic: Trigonometric Identities and Proofs

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

Key Terms and Formulas:

• $\sec x = \frac{1}{\cos x}$

• $\tan x = \frac{\sin x}{\cos x}$

• Reciprocal Identity: $\frac{1}{\sec x} = \cos x$

Step-by-Step Guidance

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

2. Combine the terms in the numerator over a common denominator.

3. Simplify the numerator and then divide by $\sin(x)$.

4. Compare your result to $\frac{1}{\sec(x)}$ and simplify if needed.

Try solving on your own before revealing the answer!

Q6. Prove the identity: $\csc\theta - \sin\theta = \cos\theta \cot\theta$

Background

Topic: Trigonometric Identities and Proofs

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

Key Terms and Formulas:

• $\csc\theta = \frac{1}{\sin\theta}$

• $\cot\theta = \frac{\cos\theta}{\sin\theta}$

Step-by-Step Guidance

1. Rewrite $\csc\theta$ and $\cot\theta$ in terms of sine and cosine.

2. Express $\csc\theta - \sin\theta$ as a single fraction.

3. Simplify the numerator and denominator.

4. Show that the result matches $\cos\theta \cot\theta$.

Try solving on your own before revealing the answer!

Q7. Prove the identity: $\csc^2\theta \tan^2\theta - 1 = \tan^2\theta$

Background

Topic: Trigonometric Identities and Proofs

This question tests your ability to manipulate and simplify trigonometric expressions using Pythagorean and reciprocal identities.

Key Terms and Formulas:

• $\csc\theta = \frac{1}{\sin\theta}$

• $\tan\theta = \frac{\sin\theta}{\cos\theta}$

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

Step-by-Step Guidance

1. Rewrite $\csc^2\theta$ and $\tan^2\theta$ in terms of sine and cosine.

2. Multiply the expressions and simplify the numerator and denominator.

3. Subtract 1 and simplify further.

4. Show that the result equals $\tan^2\theta$.

Try solving on your own before revealing the answer!

Q8. Find the exact value of $\cos 80^\circ \cos 50^\circ + \sin 80^\circ \sin 50^\circ$

Background

Topic: Sum and Difference Formulas for Cosine and Sine

This question tests your ability to recognize and apply the cosine or sine sum/difference identities.

Key Terms and Formulas:

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

• Sine Sum Formula: $\sin(A + B) = \sin A \cos B + \cos A \sin B$

Step-by-Step Guidance

1. Recognize the structure matches the cosine sum formula: $\cos(A - B)$.

2. Set $A = 80^\circ$ and $B = 50^\circ$.

3. Apply the formula: $\cos(80^\circ - 50^\circ)$.

4. Simplify the angle and write the final expression to evaluate.

Try solving on your own before revealing the answer!

Q9. Find the value of each of the following:

• a) $\sin^{-1}\left(-\frac{1}{2}\right)$

• b) $\tan^{-1} 3$

• c) $\cos^{-1}(0.5)$

• d) $\cos^{-1}(-0.5)$

Background

Topic: Inverse Trigonometric Functions

This question tests your understanding of the ranges and values of inverse trigonometric functions.

Key Terms and Formulas:

• Inverse Sine: $\sin^{-1} x$ gives the angle whose sine is $x$, in $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

• Inverse Cosine: $\cos^{-1} x$ gives the angle whose cosine is $x$, in $[0, \pi]$.

• Inverse Tangent: $\tan^{-1} x$ gives the angle whose tangent is $x$, in $(-\frac{\pi}{2}, \frac{\pi}{2})$.

Step-by-Step Guidance

1. For each part, recall the definition and range of the inverse function.

2. Determine the angle (in radians or degrees) that corresponds to the given value.

3. Check that your answer falls within the correct range for the inverse function.

4. Write the answer in the simplest exact form (e.g., $-\frac{\pi}{6}$, $\frac{\pi}{3}$, etc.).

Try solving on your own before revealing the answer!

Q10. Given $\sin\theta = -\frac{2}{5}$ and $\pi < \theta < \frac{3\pi}{2}$, find:

• a) $\sin\frac{\theta}{2}$

• b) $\cos(2\theta)$

Background

Topic: Trigonometric Values and Double/Half Angle Formulas

This question tests your ability to use double and half-angle formulas, and to determine the correct sign based on the quadrant.

Key Terms and Formulas:

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

• Double-Angle Formula for Cosine: $\cos(2\theta) = 2\cos^2\theta - 1$ or $1 - 2\sin^2\theta$

Step-by-Step Guidance

1. Draw a reference triangle for $\theta$ in the third quadrant, using $\sin\theta = -\frac{2}{5}$ to find $\cos\theta$.

2. Use the Pythagorean identity to solve for $\cos\theta$ (remember the sign in the third quadrant).

3. For part (a), apply the half-angle formula and determine the correct sign based on the quadrant of $\frac{\theta}{2}$.

4. For part (b), use the double-angle formula and substitute the value of $\sin\theta$.

Try solving on your own before revealing the answer!

Q11. Find the exact value of $\cos\left(\sin^{-1}\left(\frac{3}{4}\right) - \cos^{-1}\left(-\frac{1}{2}\right)\right)$

Background

Topic: Composition of Inverse Trigonometric Functions and Angle Difference Formulas

This question tests your ability to evaluate composite trigonometric expressions using right triangles and sum/difference formulas.

Key Terms and Formulas:

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

• Inverse Sine and Cosine: Use right triangles to find the other sides.

Step-by-Step Guidance

1. Let $A = \sin^{-1}\left(\frac{3}{4}\right)$ and $B = \cos^{-1}\left(-\frac{1}{2}\right)$.

2. Draw right triangles to find $\cos A$, $\sin A$, $\cos B$, and $\sin B$.

3. Apply the cosine difference formula using these values.

4. Simplify the resulting expression.

Try solving on your own before revealing the answer!

Q12. Find the exact value of $\sin\left[2\cos^{-1}\left(-\frac{4}{5}\right)\right]$

Background

Topic: Double-Angle Formulas and Inverse Trigonometric Functions

This question tests your ability to use the double-angle formula for sine and evaluate expressions involving inverse cosine.

Key Terms and Formulas:

• Double-Angle Formula for Sine: $\sin(2x) = 2\sin x \cos x$

• Inverse Cosine: $\cos^{-1} x$ gives the angle whose cosine is $x$.

Step-by-Step Guidance

1. Let $x = \cos^{-1}\left(-\frac{4}{5}\right)$, so $\cos x = -\frac{4}{5}$.

2. Draw a right triangle to find $\sin x$.

3. Apply the double-angle formula: $\sin(2x) = 2\sin x \cos x$.

4. Substitute the values and simplify the expression.

Try solving on your own before revealing the answer!

Q13. Solve for $0 < \theta < 2\pi$: $\sin\theta = \tan\theta$

Background

Topic: Solving Trigonometric Equations

This question tests your ability to solve equations involving multiple trigonometric functions and to find all solutions in a given interval.

Key Terms and Formulas:

• $\tan\theta = \frac{\sin\theta}{\cos\theta}$

Step-by-Step Guidance

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

2. Set $\sin\theta = \frac{\sin\theta}{\cos\theta}$ and solve for $\theta$.

3. Consider cases where $\sin\theta = 0$ and where $\cos\theta = 1$.

4. List all solutions in the interval $0 < \theta < 2\pi$.

Try solving on your own before revealing the answer!

Q14. Solve for $0 < \theta < 2\pi$: $\sin\theta = -2\sin(2\theta)$

Background

Topic: Solving Trigonometric Equations with Multiple Angles

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

Key Terms and Formulas:

• Double-Angle Formula: $\sin(2\theta) = 2\sin\theta \cos\theta$

Step-by-Step Guidance

1. Rewrite $\sin(2\theta)$ using the double-angle formula.

2. Substitute into the equation and collect like terms.

3. Factor the equation to find possible solutions for $\theta$.

4. List all solutions in the interval $0 < \theta < 2\pi$.

Try solving on your own before revealing the answer!

Q15. Solve the equation $\sin(x + x) = \ln(x)$ by graphing

Background

Topic: Graphical Solutions to Equations Involving Trigonometric and Logarithmic Functions

This question tests your ability to interpret and solve equations graphically, especially when algebraic solutions are not straightforward.

Key Terms and Formulas:

• $\sin(2x)$ is a periodic trigonometric function.

• $\ln(x)$ is the natural logarithm function, defined for $x > 0$.

Step-by-Step Guidance

1. Recognize that $\sin(x + x) = \sin(2x)$, so the equation is $\sin(2x) = \ln(x)$.

2. Sketch the graph of $y = \sin(2x)$ for $x > 0$.

3. Sketch the graph of $y = \ln(x)$ for $x > 0$ on the same axes.

4. Identify the intersection points, which represent the solutions to the equation.

Try solving on your own before revealing the answer