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 of the triangle relative to the 30° angle: opposite side is QR, adjacent side is PQ, and hypotenuse is PR.
2. Recall the definition of tangent: $\tan (\theta )=\frac{\text{opposite}}{\text{adjacent}}$.
3. For $\tan ({30}^{\circ })$, use the sides: opposite = 1, adjacent = $\sqrt{3}$.
4. Set up the expression: $\tan ({30}^{\circ })=\frac{1}{\sqrt{3}}$.
5. Rationalize the denominator: multiply numerator and denominator by $\sqrt{3}$ to get $\frac{\sqrt{3}}{3}$.

Key Concepts

Tangent Function

The tangent function, denoted as tan(θ), is a fundamental trigonometric ratio defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. For example, in triangle PQR, tan(30°) can be calculated using the lengths of the sides opposite and adjacent to the 30° angle.

Special Right Triangles

Special right triangles, specifically the 30-60-90 triangle, have known side ratios: the side opposite the 30° angle is half the hypotenuse, and the side opposite the 60° angle is √3 times the shorter leg. In triangle PQR, the sides are labeled 1 (opposite 30°), 2 (hypotenuse), and √3 (opposite 60°), which helps in easily determining trigonometric values.

Rationalizing the Denominator

Rationalizing the denominator is a mathematical 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, ensuring the expression is in a standard form.