In Exercises 47–54, use the figures to find the exact value of each trigonometric function.αcos -------2

Verified step by step guidance

1. Identify the angle \( \alpha \) in the given figure and determine its reference angle if necessary.
2. Recall the definition of the cosine function: \( \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \) for a right triangle.
3. If the figure is a right triangle, use the lengths of the sides to find \( \cos(\alpha) \).
4. To find \( \cos\left(\frac{\alpha}{2}\right) \), use the half-angle identity: \( \cos\left(\frac{\alpha}{2}\right) = \pm \sqrt{\frac{1 + \cos(\alpha)}{2}} \).
5. Determine the sign of \( \cos\left(\frac{\alpha}{2}\right) \) based on the quadrant in which \( \frac{\alpha}{2} \) lies.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. These functions are fundamental in trigonometry and are defined for right triangles, where the cosine of an angle is the ratio of the adjacent side to the hypotenuse. Understanding these functions is essential for solving problems involving angles and distances.

Cosine Function

The cosine function is one of the primary trigonometric functions, denoted as cos(θ), where θ is an angle. It is defined as the ratio of the length of the adjacent side to the hypotenuse in a right triangle. In the context of the question, finding the exact value of cos(α/2) requires knowledge of angle relationships and possibly the use of trigonometric identities.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identity, angle sum and difference identities, and double angle formulas. These identities are crucial for simplifying expressions and solving trigonometric equations, particularly when calculating values like cos(α/2).