In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.
cot 𝜋/3

Verified step by step guidance

1. Identify the angle for which you need to find the cotangent. Here, \( \pi/3 \) radians is equivalent to 60 degrees.
2. Recall that \( \cot \theta = \frac{1}{\tan \theta} \). Therefore, \( \cot 60^\circ = \frac{1}{\tan 60^\circ} \).
3. From the triangle, \( \tan 60^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{3}}{1} = \sqrt{3} \).
4. Substitute \( \tan 60^\circ \) into the cotangent formula: \( \cot 60^\circ = \frac{1}{\sqrt{3}} \).
5. Rationalize the denominator: Multiply numerator and denominator by \( \sqrt{3} \) to get \( \frac{\sqrt{3}}{3} \).

Key Concepts

Cotangent Function

The cotangent function, denoted as cot(θ), is the reciprocal of the tangent function. It is defined as the ratio of the adjacent side to the opposite side in a right triangle. For example, in triangle PQR, cot(60°) can be calculated using the lengths of the sides opposite and adjacent to the angle.

30-60-90 Triangle Properties

A 30-60-90 triangle has specific side length ratios: the side opposite the 30° angle is half the hypotenuse, and the side opposite the 60° angle is √3 times the shorter side. In triangle PQR, the sides are labeled as 1 (opposite 30°), √3 (opposite 60°), and 2 (the hypotenuse), which helps in calculating trigonometric functions.

Rationalizing the Denominator

Rationalizing the denominator is a process used to eliminate square roots or irrational numbers from the denominator of a fraction. This is often done by multiplying the numerator and denominator by a suitable value that will result in a rational number in the denominator, making the expression easier to work with.