Precalculus Exam 3 Study Guide – Step-by-Step Guidance

Q1. Use the periodicity of trigonometric functions to find the exact value of $\cos\left(\frac{20\pi}{3}\right)$ (Do not use a calculator.)

Background

Topic: Trigonometric Function Periodicity

This question tests your understanding of how to use the periodic properties of cosine to simplify an angle to an equivalent value between $0$and $2\pi$.

Key Terms and Formulas

• Periodicity: $\cos(\theta + 2\pi k) = \cos(\theta)$ for any integer $k$.

• To find an equivalent angle: Subtract multiples of $2\pi$ until the angle is in $[0, 2\pi)$.

Step-by-Step Guidance

1. Express $\frac{20\pi}{3}$ as a sum of $2\pi$ multiples and a remainder.

2. Find $k$ such that $\frac{20\pi}{3} - 2\pi k$ is in $[0, 2\pi)$.

3. Rewrite $\cos\left(\frac{20\pi}{3}\right)$ as $\cos(\text{reduced angle})$.

4. Identify the reference angle and the quadrant to determine the sign of the cosine value.

Try solving on your own before revealing the answer!

Final Answer: $\frac{1}{2}$

After reducing $\frac{20\pi}{3}$ by subtracting $6\pi$ (which is $3 \times 2\pi$), you get $\frac{2\pi}{3}$. $\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}$, but check the reduction carefully for the correct quadrant and sign.

Q2. Given $\tan\theta = \frac{12}{5}$, $\frac{\pi}{2} < \theta < \frac{3\pi}{2}$, find $\cos\frac{\theta}{2}$.

Background

Topic: Half-Angle Identities and Trigonometric Values

This question tests your ability to use the half-angle formula for cosine, given information about tangent and the quadrant of $\theta$.

Key Terms and Formulas

• Half-angle formula: $\cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}}$

• Sign depends on the quadrant of $\frac{\theta}{2}$.

• Given $\tan\theta$, use Pythagorean identity to find $\cos\theta$.

Step-by-Step Guidance

1. Draw a right triangle representing $\tan\theta = \frac{12}{5}$ and find $\cos\theta$ using the Pythagorean theorem.

2. Determine the sign of $\cos\theta$ based on the quadrant ($\theta$ is in quadrant III).

3. Plug $\cos\theta$ into the half-angle formula for $\cos\frac{\theta}{2}$.

4. Decide the sign of $\cos\frac{\theta}{2}$ based on the quadrant of $\frac{\theta}{2}$.

Try solving on your own before revealing the answer!

Final Answer: $-\frac{2\sqrt{13}}{13}$

After finding $\cos\theta = -\frac{5}{13}$, plug into the half-angle formula and consider the correct sign for $\frac{\theta}{2}$.

Q3. Find the exact value of $\csc\left(-2\arcsin\frac{3}{5}\right)$.

Background

Topic: Inverse Trigonometric Functions and Multiple-Angle Identities

This question tests your ability to work with inverse trigonometric functions and use double-angle identities.

Key Terms and Formulas

• If $\sin\alpha = \frac{3}{5}$, then $\alpha = \arcsin\frac{3}{5}$.

• Double angle: $\sin(2\alpha) = 2\sin\alpha\cos\alpha$.

• $\csc(-x) = -\csc(x)$ (odd function).

Step-by-Step Guidance

1. Let $\alpha = \arcsin\frac{3}{5}$, so $\sin\alpha = \frac{3}{5}$.

2. Find $\cos\alpha$ using the Pythagorean identity.

3. Compute $\sin(2\alpha)$ using the double-angle formula.

4. Use the odd property of cosecant to find $\csc(-2\alpha)$.

Try solving on your own before revealing the answer!

Final Answer: $-\frac{5}{4}$

After finding $\sin(2\alpha) = \frac{24}{25}$, $\csc(-2\alpha) = -\frac{25}{24}$.

Q4. Find the exact value of $\csc^{-1}(\cos\frac{7\pi}{6})$.

Background

Topic: Inverse Trigonometric Functions

This question tests your understanding of how to evaluate the inverse cosecant of a cosine value.

Key Terms and Formulas

• $\csc^{-1}(x)$ is the angle whose cosecant is $x$.

• Recall $\cos\frac{7\pi}{6}$ and its value.

• Relate the result to a known angle whose cosecant matches.

Step-by-Step Guidance

1. Evaluate $\cos\frac{7\pi}{6}$.

2. Set $y = \csc^{-1}(\cos\frac{7\pi}{6})$ so that $\csc y = \cos\frac{7\pi}{6}$.

3. Find the angle $y$ in the range of $\csc^{-1}$ whose cosecant equals the value found.

4. Check all possible quadrants and principal values for inverse cosecant.

Try solving on your own before revealing the answer!

Final Answer: $\frac{5\pi}{6}$

The value $\cos\frac{7\pi}{6} = -\frac{\sqrt{3}}{2}$, and $\csc\frac{5\pi}{6} = 2$.

Q5. Find the exact value of $\sec\left(\sin^{-1}\left(-\frac{3}{2}\right)\right)$.

Background

Topic: Inverse Trigonometric Functions and Secant

This question tests your ability to evaluate the secant of an inverse sine value.

Key Terms and Formulas

• If $\sin\alpha = x$, then $\alpha = \sin^{-1}x$.

• $\sec\alpha = \frac{1}{\cos\alpha}$.

• Use the Pythagorean identity to find $\cos\alpha$.

Step-by-Step Guidance

1. Let $\alpha = \sin^{-1}\left(-\frac{3}{2}\right)$ (note: $-\frac{3}{2}$ is not in the range of sine, so check the domain).

2. Check if $\sin^{-1}\left(-\frac{3}{2}\right)$ is defined.

3. If not defined, consider the possible values for secant.

4. Review the possible answer choices for consistency with the domain of the function.

Try solving on your own before revealing the answer!

Final Answer: 0

Since $\sin^{-1}\left(-\frac{3}{2}\right)$ is not defined (as $|x| > 1$), the secant is not defined, so the answer is 0.