Trigonometry Study Guide 3: Step-by-Step Guidance

Q1. Find csc(−𝜃) given sin(𝜃) = \frac{3}{4}

Background

Topic: Reciprocal and Negative Angle Identities

This question tests your understanding of how to find the cosecant of a negative angle using the sine value and reciprocal identities.

Key Terms and Formulas:

• Cosecant: $\csc \theta = \frac{1}{\sin \theta}$

• Negative Angle Identity: $\sin(-\theta) = -\sin \theta$

• $\csc(-\theta) = -\csc \theta$

Step-by-Step Guidance

1. Recall that $\csc(-\theta) = \frac{1}{\sin(-\theta)}$.

2. Use the negative angle identity: $\sin(-\theta) = -\sin \theta$.

3. Substitute the given value: $\sin \theta = \frac{3}{4}$, so $\sin(-\theta) = -\frac{3}{4}$.

4. Set up $\csc(-\theta) = \frac{1}{-\frac{3}{4}}$ and simplify the expression.

Try solving on your own before revealing the answer!

Final Answer: -\frac{4}{3}

$\csc(-\theta) = \frac{1}{-\frac{3}{4}} = -\frac{4}{3}$

We used the reciprocal and negative angle identities to find the exact value.

Q2. Find cos(−𝜃) given sec(𝜃) = 0.84

Background

Topic: Reciprocal and Negative Angle Identities

This question tests your ability to use reciprocal identities and properties of cosine for negative angles.

Key Terms and Formulas:

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

• Negative Angle Identity: $\cos(-\theta) = \cos \theta$

Step-by-Step Guidance

1. Recall that $\cos(-\theta) = \cos \theta$.

2. Use the reciprocal identity: $\sec \theta = \frac{1}{\cos \theta}$.

3. Given $\sec \theta = 0.84$, solve for $\cos \theta$.

4. Set up $\cos(-\theta) = \cos \theta$ and substitute the value you found.

Try solving on your own before revealing the answer!

Final Answer: 1.19

$\cos \theta = \frac{1}{0.84} \approx 1.19$

Since cosine is an even function, $\cos(-\theta) = \cos \theta$.

Q3. Find sin 𝜃 if cos 𝜃 = -\frac{3}{5} and 𝜃 \in QIII

Background

Topic: Pythagorean Identity and Quadrant Analysis

This question tests your ability to use the Pythagorean identity and determine the sign of sine based on the quadrant.

Key Terms and Formulas:

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

• Quadrant III: Both sine and cosine are negative.

Step-by-Step Guidance

1. Use the Pythagorean identity: $\sin^2 \theta = 1 - \cos^2 \theta$.

2. Substitute $\cos \theta = -\frac{3}{5}$ into the equation.

3. Calculate $\cos^2 \theta = \left(-\frac{3}{5}\right)^2 = \frac{9}{25}$.

4. Find $\sin^2 \theta = 1 - \frac{9}{25}$ and then take the square root, remembering the sign for Quadrant III.

Try solving on your own before revealing the answer!

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

$\sin \theta = -\frac{4}{5}$ because sine is negative in Quadrant III.

Q4. Use the information in problem #3 to find tan 𝜃

Background

Topic: Tangent Identity and Quadrant Analysis

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

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Recall $\tan \theta = \frac{\sin \theta}{\cos \theta}$.

2. Use $\sin \theta = -\frac{4}{5}$ and $\cos \theta = -\frac{3}{5}$ from the previous problem.

3. Set up $\tan \theta = \frac{-\frac{4}{5}}{-\frac{3}{5}}$ and simplify.

Try solving on your own before revealing the answer!

Final Answer: \frac{4}{3}

$\tan \theta = \frac{4}{3}$ because both sine and cosine are negative, so tangent is positive in Quadrant III.

Q5. Simplify: \frac{\cot x}{\csc x}

Background

Topic: Simplifying Trigonometric Expressions

This question tests your ability to simplify trigonometric expressions using basic identities.

Key Terms and Formulas:

• Cotangent: $\cot x = \frac{\cos x}{\sin x}$

• Cosecant: $\csc x = \frac{1}{\sin x}$

Step-by-Step Guidance

1. Write $\cot x$ and $\csc x$ in terms of sine and cosine.

2. Set up $\frac{\cot x}{\csc x} = \frac{\frac{\cos x}{\sin x}}{\frac{1}{\sin x}}$.

3. Simplify the expression by multiplying numerator and denominator.

Try solving on your own before revealing the answer!

Final Answer: \cos x

After simplification, $\frac{\cot x}{\csc x} = \cos x$.

Q6. Find an exact value for: cos(30° −𝜃)

Background

Topic: Sum and Difference Identities

This question tests your ability to use the cosine difference identity to find an exact value.

Key Terms and Formulas:

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

• Values for $\cos 30^\circ = \frac{\sqrt{3}}{2}$ and $\sin 30^\circ = \frac{1}{2}$

Step-by-Step Guidance

1. Apply the cosine difference identity: $\cos(30^\circ - \theta) = \cos 30^\circ \cos \theta + \sin 30^\circ \sin \theta$.

2. Substitute $\cos 30^\circ = \frac{\sqrt{3}}{2}$ and $\sin 30^\circ = \frac{1}{2}$.

3. Set up the expression: $\frac{\sqrt{3}}{2} \cos \theta + \frac{1}{2} \sin \theta$.

Try solving on your own before revealing the answer!

Final Answer: \frac{\sqrt{3}}{2} \cos \theta + \frac{1}{2} \sin \theta

We used the sum and difference identity and substituted the known values for 30°.

Unit Circle Reference

The unit circle is a fundamental tool in trigonometry for finding exact values of sine, cosine, and tangent at standard angles. It helps visualize the relationships between angles and their trigonometric values.