Precalculus Trigonometric Functions and Identities Study Guide

Q1. Find the five remaining trigonometric functions of \( \alpha \) given \( \tan \alpha = -\frac{1}{4} \), \( \alpha \) in quadrant IV.

Background

Topic: Trigonometric Functions and Reference Triangles

This question tests your ability to use the value of one trigonometric function and quadrant information to find the other five trigonometric functions for a given angle.

Key Terms and Formulas:

• \( \tan \alpha = \frac{\text{opposite}}{\text{adjacent}} \)

• \( \sin \alpha = \frac{\text{opposite}}{\text{hypotenuse}} \)

• \( \cos \alpha = \frac{\text{adjacent}}{\text{hypotenuse}} \)

• Pythagorean Theorem: \( a^2 + b^2 = c^2 \)

• uadrant IV: sine and cosecant are negative, cosine and secant are positive, tangent and cotangent are negative.

Step-by-Step Guidance

1. Assign values to the sides of a right triangle based on \( \tan \alpha = -\frac{1}{4} \). Let the opposite side be \( -1 \) (since tangent is negative in quadrant IV) and the adjacent side be \( 4 \).

2. Use the Pythagorean theorem to find the hypotenuse: \( \sqrt{(-1)^2 + 4^2} \).

3. Write expressions for \( \sin \alpha \), \( \cos \alpha \), \( \csc \alpha \), \( \sec \alpha \), and \( \cot \alpha \) using the triangle sides and the quadrant information to determine the correct signs.

4. Remember to rationalize denominators and simplify all radicals in your answers.

Try solving on your own before revealing the answer!

Final Answer:

\( \cot \alpha = -4 \), \( \sin \alpha = -\frac{1}{\sqrt{17}} \), \( \cos \alpha = \frac{4}{\sqrt{17}} \), \( \csc \alpha = -\frac{\sqrt{17}}{1} \), \( \sec \alpha = \frac{\sqrt{17}}{4} \)

All values are determined using the reference triangle and the signs for quadrant IV.

Q2. Write each expression in terms of sine and cosine, then simplify: \( \sec^2(-\theta) - \cos^2(-\theta) - \tan^2(-\theta) \)

Background

Topic: Trigonometric Identities and Simplification

This question tests your ability to rewrite trigonometric expressions using sine and cosine, and to simplify using identities.

Key Terms and Formulas:

• \( \sec \theta = \frac{1}{\cos \theta} \)

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

• Even-odd identities: \( \cos(-\theta) = \cos \theta \), \( \sin(-\theta) = -\sin \theta \)

Step-by-Step Guidance

1. Rewrite each function in terms of sine and cosine, applying the even-odd identities as needed.

2. \( \sec^2(-\theta) = \frac{1}{\cos^2(-\theta)} = \frac{1}{\cos^2 \theta} \)

3. \( \cos^2(-\theta) = \cos^2 \theta \)

4. \( \tan^2(-\theta) = \left(\frac{\sin(-\theta)}{\cos(-\theta)}\right)^2 = \left(\frac{-\sin \theta}{\cos \theta}\right)^2 = \frac{\sin^2 \theta}{\cos^2 \theta} \)

5. Combine all terms over a common denominator and simplify as much as possible, but stop before the final simplification.

Try solving on your own before revealing the answer!

Final Answer:

\( \sec^2(-\theta) - \cos^2(-\theta) - \tan^2(-\theta) = 1 \)

This follows from the Pythagorean identity and simplification.

Q3. Verify the identity: \( \frac{1+\cos x}{1-\cos x} \cdot \frac{1-\cos x}{1+\cos x} = 4 \cot x \csc x \)

Background

Topic: Trigonometric Identities and Proofs

This question tests your ability to manipulate and verify trigonometric identities by transforming one side to match the other.

Key Terms and Formulas:

• \( \cot x = \frac{\cos x}{\sin x} \)

• \( \csc x = \frac{1}{\sin x} \)

• Pythagorean identities: \( 1 - \cos^2 x = \sin^2 x \)

Step-by-Step Guidance

1. Start with the left side: \( \frac{1+\cos x}{1-\cos x} \cdot \frac{1-\cos x}{1+\cos x} \).

2. Multiply the numerators and denominators to simplify the expression.

3. Look for opportunities to use Pythagorean identities to rewrite the result in terms of sine and cosine.

4. Continue simplifying until the expression resembles the right side, but stop before the final transformation.

Try solving on your own before revealing the answer!

Final Answer:

\( \frac{1+\cos x}{1-\cos x} \cdot \frac{1-\cos x}{1+\cos x} = 4 \cot x \csc x \)

Both sides are equivalent after simplification using trigonometric identities.

Final Answer:

\( \cos \left( \frac{7\pi}{12} \right) = \frac{\sqrt{6} - \sqrt{2}}{4} \)

This uses the sum identity and exact values for the standard angles.