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.
cos 30°

Verified step by step guidance

1. Identify the sides of the triangle relative to the 30° angle: the hypotenuse is PR = 2, the adjacent side is QR = 1, and the opposite side is PQ = \( \sqrt{3} \).
2. Recall the definition of cosine for an angle in a right triangle: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
3. Substitute the known values into the cosine formula: \( \cos(30°) = \frac{QR}{PR} = \frac{1}{2} \).
4. Since the value is already simplified, there is no need to rationalize the denominator.
5. Conclude that the expression for \( \cos(30°) \) is \( \frac{1}{2} \).

Key Concepts

Trigonometric Ratios

Trigonometric ratios are relationships between the angles and sides of a right triangle. The primary ratios include sine, cosine, and tangent, defined as the ratios of the lengths of the sides opposite, adjacent, and hypotenuse to a given angle. For example, in triangle PQR, cos(30°) is calculated as the length of the adjacent side (QR) over the hypotenuse (PR).

Special Right Triangles

Special right triangles, specifically the 30-60-90 triangle, have known side ratios that simplify calculations. In a 30-60-90 triangle, the sides opposite the 30°, 60°, and 90° angles are in the ratio 1:√3:2. This means that if the shortest side is 1, the longer leg is √3, and the hypotenuse is 2, making it easier to find trigonometric values without complex calculations.

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. For example, if the expression involves a square root in the denominator, multiplying by that square root can help simplify the expression.