In Exercises 49–59, find the exact value of each expression. Do not use a calculator. cot(-210°)

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Recall the definition of cotangent: \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). To find \(\cot(-210^\circ)\), we first need to understand the angle \(-210^\circ\) in terms of its position on the unit circle.

Convert the negative angle to a positive coterminal angle by adding \(360^\circ\): \(-210^\circ + 360^\circ = 150^\circ\). So, \(\cot(-210^\circ) = \cot(150^\circ)\).

Identify the reference angle for \(150^\circ\). Since \(150^\circ\) is in the second quadrant, the reference angle is \(180^\circ - 150^\circ = 30^\circ\).

Determine the signs of sine and cosine in the second quadrant: sine is positive and cosine is negative. Use the reference angle to find \(\sin 150^\circ = \sin 30^\circ\) and \(\cos 150^\circ = -\cos 30^\circ\).

Calculate \(\cot 150^\circ\) using the ratio \(\cot 150^\circ = \frac{\cos 150^\circ}{\sin 150^\circ} = \frac{-\cos 30^\circ}{\sin 30^\circ}\). Substitute the exact values for \(\sin 30^\circ\) and \(\cos 30^\circ\) to express the exact value.

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Cotangent Function and Its Definition

Cotangent is a trigonometric function defined as the ratio of the adjacent side to the opposite side in a right triangle, or equivalently, cot(θ) = cos(θ)/sin(θ). Understanding this ratio is essential for evaluating cotangent values without a calculator.

Reference Angles and Angle Reduction

To find the exact value of trigonometric functions for angles outside the first quadrant, use reference angles by reducing the given angle to an acute angle within 0° to 90°. This involves adding or subtracting full rotations (360°) or using symmetry properties of the unit circle.

Sign of Trigonometric Functions in Different Quadrants

The sign of cotangent depends on the quadrant in which the angle lies. Since cotangent is cos(θ)/sin(θ), knowing the signs of sine and cosine in each quadrant helps determine whether cotangent is positive or negative.