Trigonometry Exam & Worksheet Step-by-Step Guidance

Q1. Find $\sin \theta$ given $\cot \theta = -\frac{4}{5}$ and $\theta$ is in quadrant IV.

Background

Topic: Trigonometric Functions & Quadrant Analysis

This question tests your understanding of cotangent, sine, and the signs of trigonometric functions in different quadrants.

Key Terms and Formulas:

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

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

• Quadrant IV: $\cos \theta > 0$, $\sin \theta < 0$

Step-by-Step Guidance

1. Express $\cot \theta = -\frac{4}{5}$ as $\frac{\cos \theta}{\sin \theta} = -\frac{4}{5}$.

2. Let $\cos \theta = -4k$ and $\sin \theta = 5k$ for some $k$ (the negative sign ensures the correct sign in quadrant IV).

3. Use the Pythagorean identity to solve for $k$: $\sin^2 \theta + \cos^2 \theta = 1$ Substitute the expressions for $\sin \theta$ and $\cos \theta$.

4. Solve for $k$ and then find $\sin \theta$ using the correct sign for quadrant IV.

Try solving on your own before revealing the answer!

Q2. Find $\sin \theta$ given $\tan \theta = -\frac{10}{3}$ and $\cos \theta > 0$.

Background

Topic: Trigonometric Ratios & Quadrant Analysis

This question tests your ability to use tangent and cosine information to find sine, considering the correct quadrant.

Key Terms and Formulas:

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

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

• Sign analysis: $\tan \theta < 0$ and $\cos \theta > 0$ implies $\sin \theta < 0$ (Quadrant IV)

Step-by-Step Guidance

1. Write $\tan \theta = -\frac{10}{3} = \frac{\sin \theta}{\cos \theta}$.

2. Let $\sin \theta = -10k$ and $\cos \theta = 3k$ for some $k$ (negative sign for sine in quadrant IV).

3. Apply the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$ Substitute the expressions for $\sin \theta$ and $\cos \theta$.

4. Solve for $k$ and then find $\sin \theta$.

Try solving on your own before revealing the answer!

Q3. Identify the basic trigonometric function graphed and determine whether it is even or odd.

Background

Topic: Graphs of Trigonometric Functions & Even/Odd Properties

This question tests your ability to recognize the graph of a basic trig function and classify it as even or odd.

Key Terms and Formulas:

• Even function: $f(-x) = f(x)$ (e.g., $\cos x$, $\sec x$)

• Odd function: $f(-x) = -f(x)$ (e.g., $\sin x$, $\tan x$, $\cot x$, $\csc x$)

Step-by-Step Guidance

1. Examine the graph and compare its shape to the standard graphs of $\sin x$, $\cos x$, $\tan x$, etc.

2. Determine if the function is symmetric about the y-axis (even) or origin (odd).

3. Match the graph to the correct function and classify its parity.

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Q4. Find the remaining five trigonometric functions of $\theta$, given $\cot \theta = \frac{5}{12}$ and $\sin \theta < 0$.

Background

Topic: Trigonometric Ratios & Signs in Quadrants

This question tests your ability to find all trig functions given one and the sign of sine.

Key Terms and Formulas:

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

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

• Signs: $\sin \theta < 0$ (so $\theta$ is in quadrant III or IV)

Step-by-Step Guidance

1. Let $\cot \theta = \frac{5}{12}$, so $\cos \theta = 5k$, $\sin \theta = 12k$ for some $k$.

2. Since $\sin \theta < 0$, both $\sin \theta$ and $\cos \theta$ must be negative (quadrant III).

3. Use the Pythagorean identity to solve for $k$.

4. Find all six trig functions using the values for $\sin \theta$ and $\cos \theta$.

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Q5. Write $-\sin^2(-\theta) + \sec^2(-\theta) - \cos^2(-\theta)$ in terms of sine and cosine, and simplify so that no quotients appear and all functions are of $\theta$ only.

Background

Topic: Trigonometric Identities (Even-Odd, Reciprocal, Pythagorean)

This question tests your ability to use even-odd identities and rewrite expressions in terms of sine and cosine.

Key Terms and Formulas:

• Even-Odd Identities: $\sin(-\theta) = -\sin \theta$, $\cos(-\theta) = \cos \theta$, $\sec(-\theta) = \sec \theta$

• $\sec \theta = \frac{1}{\cos \theta}$

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

Step-by-Step Guidance

1. Apply even-odd identities to each term to rewrite them in terms of $\theta$ (not $-\theta$).

2. Rewrite $\sec^2 \theta$ in terms of $\cos^2 \theta$.

3. Combine like terms and use the Pythagorean identity to simplify the expression.

Try solving on your own before revealing the answer!