In Exercises 39–46, use a half-angle formula to find the exact value of each expression. cos 22.5°

Key Concepts

Half-Angle Formulas

Half-angle formulas are trigonometric identities that express the sine and cosine of half an angle in terms of the sine and cosine of the original angle. For cosine, the formula is cos(θ/2) = ±√((1 + cos(θ))/2). These formulas are essential for simplifying expressions involving angles that are not standard, such as 22.5°, which is half of 45°.

Trigonometric Values of Special Angles

Certain angles, such as 0°, 30°, 45°, 60°, and 90°, have known sine and cosine values that are often used in calculations. For example, cos(45°) = √2/2. Knowing these values allows for easier computation when applying half-angle formulas, as they provide the necessary input for finding the cosine of 22.5°.

Quadrants and Sign of Trigonometric Functions

The sign of trigonometric functions depends on the quadrant in which the angle lies. For angles between 0° and 90°, both sine and cosine are positive. Since 22.5° is in the first quadrant, the cosine value derived from the half-angle formula will also be positive, which is crucial for determining the exact value of cos(22.5°).