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.

tan 30°

Verified step by step guidance

1. Identify the sides relative to the 30° angle in the triangle. The side opposite the 30° angle is QR, which has length 1. The side adjacent to the 30° angle is PQ, which has length \( \sqrt{3} \).
2. Recall the definition of the tangent function: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). For \( \theta = 30^\circ \), this becomes \( \tan 30^\circ = \frac{\text{opposite side}}{\text{adjacent side}} \).
3. Substitute the lengths of the sides into the tangent ratio: \( \tan 30^\circ = \frac{1}{\sqrt{3}} \).
4. Since the denominator contains a square root, rationalize the denominator by multiplying numerator and denominator by \( \sqrt{3} \): \( \tan 30^\circ = \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \).
5. Thus, the expression for \( \tan 30^\circ \) is simplified and rationalized as \( \frac{\sqrt{3}}{3} \).

Key Concepts

Right Triangle Trigonometric Ratios

In a right triangle, the tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to it. This fundamental definition helps in calculating the tangent value using the triangle's side lengths.

Special Angles in Trigonometry (30°-60°-90° Triangle)

A 30°-60°-90° triangle has side lengths in a fixed ratio: the side opposite 30° is half the hypotenuse, the side opposite 60° is √3 times the shorter leg, and the hypotenuse is twice the shorter leg. This ratio simplifies finding trigonometric values for these angles.

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any square roots from the denominator of a fraction by multiplying numerator and denominator by a suitable radical. This process simplifies expressions and is often required for final answers in trigonometry.