Step-by-Step Guidance for Precalculus Trigonometric Questions

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

Background

Topic: Trigonometric Functions and Quadrants

This question tests your ability to use a given trigonometric ratio and quadrant information to find all six trigonometric functions for an angle.

Key Terms and Formulas:

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

• \( \cot \alpha = \frac{1}{\tan \alpha} \)

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

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

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

• \( \csc \alpha = \frac{1}{\sin \alpha} \)

• Pythagorean identity: \( \sin^2 \alpha + \cos^2 \alpha = 1 \)

Step-by-Step Guidance

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

2. Find the hypotenuse using the Pythagorean theorem: \( \text{hypotenuse} = \sqrt{(-1)^2 + 4^2} \).

3. Write expressions for \( \sin \alpha \), \( \cos \alpha \), \( \sec \alpha \), \( \csc \alpha \), and \( \cot \alpha \) using the sides of the triangle.

4. Remember the signs of the functions in quadrant IV: sine is negative, cosine is positive, tangent is negative.

Try solving on your own before revealing the answer!

Final Answer:

\( \cot \alpha = -4 \)

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

All values are rationalized and signs are correct for quadrant IV.

Q2. Simplify \( \sec^2(-\theta) - \cos^2(-\theta) - \tan^2(-\theta) \) in terms of sine and cosine, so that no quotients appear and all functions are of \( \theta \) only.

Background

Topic: Trigonometric Identities and Simplification

This question tests your ability to rewrite trigonometric expressions using fundamental identities and to express everything in terms of sine and cosine.

Key Terms and Formulas:

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

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

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

Step-by-Step Guidance

1. Rewrite each function in terms of sine and cosine, using the even-odd identities for negative angles.

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

3. Express \( \tan^2 \theta \) as \( \frac{\sin^2 \theta}{\cos^2 \theta} \).

4. Combine all terms over a common denominator to eliminate quotients.

Try solving on your own before revealing the answer!

Final Answer:

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

After simplification, the expression reduces to 1 using the Pythagorean identity.

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:

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

• \( \csc x = \frac{1}{\sin x} \)

• \( 1 - \cos^2 x = \sin^2 x \)

Step-by-Step Guidance

1. Multiply the two fractions on the left side. Notice that the numerators and denominators will cancel.

2. Simplify the resulting expression and rewrite in terms of sine and cosine.

3. Use the Pythagorean identity to substitute for \( 1 - \cos^2 x \).

4. Express the result in terms of \( \cot x \) and \( \csc x \).

Try solving on your own before revealing the answer!

Final Answer:

The equation is an identity: \( 4 \cot x \csc x \).

After simplification, both sides are equal using trigonometric identities.

Q4. Find the exact value of \( \cos \left( \frac{7\pi}{12} \right) \).

Background

Topic: Exact Values of Trigonometric Functions

This question tests your ability to use sum and difference formulas to find the exact value of a trigonometric function for a non-standard angle.

Key Terms and Formulas:

• Sum formula: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)

• Choose \( a \) and \( b \) so that their sum is \( \frac{7\pi}{12} \) and both are angles with known exact values.

Step-by-Step Guidance

1. Express \( \frac{7\pi}{12} \) as a sum of two angles with known values, such as \( \frac{3\pi}{4} \) and \( \frac{\pi}{3} \).

2. Apply the sum formula for cosine.

3. Substitute the exact values for \( \cos \frac{3\pi}{4} \), \( \cos \frac{\pi}{3} \), \( \sin \frac{3\pi}{4} \), and \( \sin \frac{\pi}{3} \).

4. Simplify the expression, rationalizing denominators as needed.

Try solving on your own before revealing the answer!

Final Answer:

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

Used sum formula and exact values for special angles.

Q5. Use the cosine of a sum and difference identities to find \( \cos(s + t) \), given \( \cos s = -\frac{3}{5} \) and \( \sin t = -\frac{1}{5} \), with \( s \) and \( t \) in quadrant III.

Background

Topic: Sum and Difference Formulas for Cosine

This question tests your ability to use sum and difference identities and quadrant information to find exact values.

Key Terms and Formulas:

• \( \cos(s + t) = \cos s \cos t - \sin s \sin t \)

• Find \( \sin s \) and \( \cos t \) using Pythagorean identities and quadrant signs.

Step-by-Step Guidance

1. Find \( \sin s \) using \( \sin^2 s + \cos^2 s = 1 \) and remember the sign in quadrant III.

2. Find \( \cos t \) using \( \sin^2 t + \cos^2 t = 1 \) and remember the sign in quadrant III.

3. Plug all values into the sum formula for cosine.

4. Simplify the expression, rationalizing denominators as needed.

Try solving on your own before revealing the answer!

Final Answer:

\( \cos(s + t) = -\frac{12}{25} \)

All values are negative in quadrant III, and denominators are rationalized.