Give the exact value of each expression. See Example 5.cos 30°

Verified step by step guidance

1. Step 1: Recognize that 30° is a special angle in trigonometry, often associated with the 30-60-90 triangle.
2. Step 2: Recall the properties of a 30-60-90 triangle, where the sides are in the ratio 1 : \( \sqrt{3} \) : 2.
3. Step 3: Identify that in a 30-60-90 triangle, the cosine of 30° is the ratio of the adjacent side to the hypotenuse.
4. Step 4: Use the side ratio to find \( \cos 30° = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2} \).
5. Step 5: Conclude that the exact value of \( \cos 30° \) is \( \frac{\sqrt{3}}{2} \).

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, which correspond to the ratios of the lengths of the sides opposite, adjacent, and hypotenuse to a given angle. Understanding these ratios is essential for calculating the values of trigonometric functions for specific angles.

Special Angles

Special angles in trigonometry refer to specific angles such as 0°, 30°, 45°, 60°, and 90°, for which the sine, cosine, and tangent values are commonly known and can be derived from geometric principles. For example, cos 30° equals √3/2. Familiarity with these angles allows for quick calculations and a deeper understanding of trigonometric functions.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a fundamental tool in trigonometry that helps define the sine and cosine functions for all angles, not just those in right triangles. By using the unit circle, one can easily find the exact values of trigonometric functions for any angle, including those beyond 0° to 360°.