Use reference angles to find the exact value of each expression. Do not use a calculator. cot 19𝜋/6

Verified step by step guidance

1. First, recognize that the angle given is in radians: \(19\pi/6\). Since the trigonometric functions are periodic, reduce the angle to an equivalent angle between \(0\) and \(2\pi\) by subtracting multiples of \(2\pi\).
2. Calculate how many full rotations of \(2\pi\) fit into \(19\pi/6\). Since \(2\pi = 12\pi/6\), subtract \(12\pi/6\) from \(19\pi/6\) to get the reference angle within one full rotation: \(19\pi/6 - 12\pi/6 = 7\pi/6\).
3. Identify the quadrant where the angle \(7\pi/6\) lies. Since \(\pi = 6\pi/6\), \(7\pi/6\) is just past \(\pi\), so it lies in the third quadrant.
4. Find the reference angle for \(7\pi/6\) by subtracting \(\pi\): Reference angle \(= 7\pi/6 - \pi = 7\pi/6 - 6\pi/6 = \pi/6\).
5. Use the reference angle \(\pi/6\) to find \(\cot(\pi/6)\), then determine the sign of \(\cot(7\pi/6)\) based on the quadrant (third quadrant). Recall that \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\) and that both sine and cosine are negative in the third quadrant, so cotangent is positive there.

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 90° (or 0 and π/2 radians). Using reference angles allows you 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 and known trigonometric values.

Angle Reduction and Coterminal Angles

Angles larger than 2π radians can be reduced by subtracting multiples of 2π to find a coterminal angle within one full rotation. This simplification helps identify the reference angle and the quadrant, which determines the sign of the trigonometric function.