Precalculus Final Review: Step-by-Step Study Guidance

Q1. Find the exact values for θ given the following trigonometric equations:

• (a) $\tan \theta = 1$

• (b) $\theta = \arctan(1)$

• (c) $\theta = \arcsin 0$

• (d) $\sin \theta = 0$

• (e) $\cos \theta = \frac{1}{2}$

• (f) $\theta = \arccos \frac{1}{2}$

• (g) $\theta = \arcsin \left(-\frac{\sqrt{3}}{2}\right)$

• (h) $\sin \theta = -\frac{\sqrt{3}}{2}$

• (i) $\theta = \arctan \sqrt{3}$

• (j) $\tan \theta = \sqrt{3}$

Background

Topic: Inverse Trigonometric Functions and Exact Trigonometric Values

This question tests your understanding of how to find exact angle values (in radians) for given trigonometric equations, using both the unit circle and inverse trigonometric functions.

Key Terms and Formulas

• Unit Circle: A circle of radius 1 centered at the origin, used to define trigonometric functions for all real numbers.

• Inverse Trigonometric Functions: $\arcsin(x)$, $\arccos(x)$, $\arctan(x)$ return the angle whose sine, cosine, or tangent is $x$.

• Reference Angles: The acute angle formed by the terminal side of an angle and the x-axis.

• Key Values: Know the sine, cosine, and tangent values for $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$, and their negatives.

Step-by-Step Guidance

1. Recall the definition of the trigonometric function or inverse function in each part. For example, $\tan \theta = 1$ asks for all angles where the tangent is 1, while $\theta = \arctan(1)$ asks for the principal value whose tangent is 1.

2. Use the unit circle to identify the angle(s) that satisfy the equation. For $\tan \theta = 1$, look for points where the y-coordinate divided by the x-coordinate equals 1.

3. For inverse functions, remember their principal value ranges:

   • $\arcsin(x)$: $[-\frac{\pi}{2}, \frac{\pi}{2}]$

   • $\arccos(x)$: $[0, \pi]$

   • $\arctan(x)$: $(-\frac{\pi}{2}, \frac{\pi}{2})$

4. For equations like $\sin \theta = 0$ or $\cos \theta = \frac{1}{2}$, determine all solutions in $[0, 2\pi)$ by considering the periodicity and symmetry of the trigonometric functions.

5. Write the solutions in radians, and check if there are multiple solutions within the interval $[0, 2\pi)$.

Try solving on your own before revealing the answer!