Match each expression in Column I with its value in Column II.
10. cos 67.5°

Verified step by step guidance

1. Recognize that 67.5° is half of 135°, so we can use the half-angle identity for cosine.
2. The half-angle identity for cosine is: \( \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} \).
3. Substitute \( \theta = 135° \) into the identity: \( \cos(67.5°) = \pm \sqrt{\frac{1 + \cos(135°)}{2}} \).
4. Calculate \( \cos(135°) \) using the unit circle or known values: \( \cos(135°) = -\frac{\sqrt{2}}{2} \).
5. Substitute \( \cos(135°) = -\frac{\sqrt{2}}{2} \) into the half-angle formula and simplify to find \( \cos(67.5°) \).

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function defined for an angle in a right triangle as the ratio of the length of the adjacent side to the hypotenuse. It is also represented on the unit circle, where the cosine of an angle corresponds to the x-coordinate of the point on the circle. Understanding the cosine function is essential for evaluating angles and solving problems involving right triangles.

Angle Measurement

Angles can be measured in degrees or radians, with degrees being a more common unit in basic trigonometry. The angle 67.5° is a specific angle that can be expressed in radians as π/4 radians. Recognizing how to convert between these units and understanding their implications in trigonometric calculations is crucial for solving problems accurately.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. For example, the half-angle identities can be used to find the cosine of angles like 67.5° by relating it to known values. Familiarity with these identities allows for simplification and evaluation of trigonometric expressions effectively.