In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. cot(7𝜋/4)

Verified step by step guidance

1. First, recognize that the angle given is \( \frac{7\pi}{4} \), which is in radians. Since \( 2\pi \) radians correspond to a full circle, identify the quadrant where \( \frac{7\pi}{4} \) lies by comparing it to \( 2\pi \).
2. Find the reference angle for \( \frac{7\pi}{4} \) by subtracting it from \( 2\pi \): \( \text{Reference angle} = 2\pi - \frac{7\pi}{4} \).
3. Recall that \( \cot \theta = \frac{1}{\tan \theta} \), so the value of \( \cot \frac{7\pi}{4} \) will be the reciprocal of \( \tan \frac{7\pi}{4} \).
4. Determine the sign of \( \cot \frac{7\pi}{4} \) based on the quadrant where \( \frac{7\pi}{4} \) lies. Remember that cotangent is positive in quadrants where tangent is positive and negative where tangent is negative.
5. Use the reference angle to find the exact value of \( \tan \) at that angle, then take its reciprocal to find \( \cot \frac{7\pi}{4} \), applying the correct sign from the previous step.

Key Concepts

Reference Angles

A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It helps simplify trigonometric calculations by relating any angle to an angle between 0 and π/2. Using reference angles allows us to find exact trigonometric values without a calculator.

Cotangent Function

Cotangent is the reciprocal of the tangent function, defined as cot(θ) = 1/tan(θ) = cos(θ)/sin(θ). Understanding cotangent's relationship to sine and cosine is essential for evaluating its exact value, especially when using reference angles.

Angle Reduction and Quadrants

Angles greater than 2π or negative angles can be reduced by subtracting or adding multiples of 2π to find a coterminal angle within one full rotation. Knowing the quadrant of the angle helps determine the sign of the trigonometric function based on the ASTC (All Students Take Calculus) rule.