Trigonometry Test 2 Review – Step-by-Step Study Guidance

Q1a. Find the exact radian value for arccos(0.5).

Background

Topic: Inverse Trigonometric Functions

This question tests your understanding of the arccosine (inverse cosine) function and your ability to find exact values in radians for common trigonometric ratios.

Key Terms and Formulas:

• arccos(x): The angle whose cosine is x, with range $0 \leq \theta \leq \pi$.

• Common exact values: $\cos(0) = 1$, $\cos\left(\frac{\pi}{3}\right) = 0.5$, $\cos\left(\frac{\pi}{2}\right) = 0$.

Step-by-Step Guidance

1. Recall the definition: $\arccos(x)$ gives the angle $\theta$ in $[0, \pi]$ such that $\cos(\theta) = x$.

2. Set up the equation: $\cos(\theta) = 0.5$.

3. Think about the unit circle and which angle(s) in $[0, \pi]$ have a cosine of $0.5$.

4. Recall the common reference angles and their cosine values.

Try solving on your own before revealing the answer!

Q1b. Find the exact radian value for $\cos^{-1}(-0.5)$.

Background

Topic: Inverse Trigonometric Functions

This question asks you to find the angle in radians whose cosine is $-0.5$, within the principal range of the arccosine function.

Key Terms and Formulas:

• $\arccos(x)$: Returns an angle $\theta$ in $[0, \pi]$ such that $\cos(\theta) = x$.

• Common values: $\cos(\pi) = -1$, $\cos\left(\frac{2\pi}{3}\right) = -0.5$.

Step-by-Step Guidance

1. Set $\cos(\theta) = -0.5$ and recall the range for $\arccos$ is $[0, \pi]$.

2. Think about which angle in this range has a cosine of $-0.5$.

3. Use your knowledge of the unit circle and reference angles to identify the correct value.

Try solving on your own before revealing the answer!

Q1c. Find the exact radian value for $\tan^{-1}(1)$.

Background

Topic: Inverse Trigonometric Functions

This question tests your understanding of the arctangent function and its principal values.

Key Terms and Formulas:

• $\arctan(x)$: Returns an angle $\theta$ in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ such that $\tan(\theta) = x$.

• Common values: $\tan\left(\frac{\pi}{4}\right) = 1$.

Step-by-Step Guidance

1. Set $\tan(\theta) = 1$ and recall the range for $\arctan$ is $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

2. Think about which angle in this range has a tangent of $1$.

3. Recall the common reference angles for tangent values.

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Q1d. Find the exact radian value for $\arctan(-1)$.

Background

Topic: Inverse Trigonometric Functions

This question asks for the angle in the principal range of arctangent whose tangent is $-1$.

Key Terms and Formulas:

• $\arctan(x)$: Returns an angle $\theta$ in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ such that $\tan(\theta) = x$.

• Common values: $\tan\left(-\frac{\pi}{4}\right) = -1$.

Step-by-Step Guidance

1. Set $\tan(\theta) = -1$ and recall the range for $\arctan$.

2. Think about which angle in this range has a tangent of $-1$.

3. Recall the sign of tangent in different quadrants and the reference angle.

Try solving on your own before revealing the answer!

Q1e. Find the exact radian value for $\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)$.

Background

Topic: Inverse Trigonometric Functions

This question tests your ability to find the principal value of the arcsine function for a common ratio.

Key Terms and Formulas:

• $\arcsin(x)$: Returns an angle $\theta$ in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ such that $\sin(\theta) = x$.

• Common values: $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$.

Step-by-Step Guidance

1. Set $\sin(\theta) = \frac{\sqrt{3}}{2}$ and recall the range for $\arcsin$.

2. Think about which angle in this range has a sine of $\frac{\sqrt{3}}{2}$.

3. Recall the common reference angles for sine values.

Try solving on your own before revealing the answer!

Q1f. Find the exact radian value for $\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)$.

Background

Topic: Inverse Trigonometric Functions

This question asks for the principal value of arcsine for a negative common ratio.

Key Terms and Formulas:

• $\arcsin(x)$: Returns an angle $\theta$ in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ such that $\sin(\theta) = x$.

• Odd function property: $\arcsin(-x) = -\arcsin(x)$.

Step-by-Step Guidance

1. Set $\sin(\theta) = -\frac{\sqrt{3}}{2}$ and recall the range for $\arcsin$.

2. Think about which angle in this range has a sine of $-\frac{\sqrt{3}}{2}$.

3. Use the odd function property to relate this to the positive case.

Try solving on your own before revealing the answer!

Q1g. Find the exact value of $\sin(\sec^{-1}(-\sqrt{2}))$.

Background

Topic: Composition of Inverse Trigonometric Functions

This question tests your ability to evaluate a trigonometric function of an inverse trigonometric function, using right triangle relationships and the definitions of secant and sine.

Key Terms and Formulas:

• $\sec^{-1}(x)$: Returns an angle $\theta$ such that $\sec(\theta) = x$ and $\theta$ is in $[0, \pi]$, $\theta \neq \frac{\pi}{2}$.

• $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$.

• Recall: $\sec(\theta) = \frac{1}{\cos(\theta)}$.

Step-by-Step Guidance

1. Let $\theta = \sec^{-1}(-\sqrt{2})$, so $\sec(\theta) = -\sqrt{2}$.

2. Recall that $\sec(\theta) = \frac{1}{\cos(\theta)}$, so $\cos(\theta) = -\frac{1}{\sqrt{2}}$.

3. Draw a right triangle (or use the unit circle) with $\cos(\theta) = -\frac{1}{\sqrt{2}}$ to find $\sin(\theta)$.

4. Use the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ to solve for $\sin(\theta)$.

Try solving on your own before revealing the answer!