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

Verified step by step guidance

1. Start by recognizing that 22.5° is half of 45°, so you can use the half-angle formula for cosine: \( \cos \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 + \cos \theta}{2}} \).
2. Substitute \( \theta = 45° \) into the formula: \( \cos 22.5° = \pm \sqrt{\frac{1 + \cos 45°}{2}} \).
3. Recall that \( \cos 45° = \frac{\sqrt{2}}{2} \). Substitute this value into the equation: \( \cos 22.5° = \pm \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} \).
4. Simplify the expression inside the square root: \( \frac{1 + \frac{\sqrt{2}}{2}}{2} = \frac{2 + \sqrt{2}}{4} \).
5. Finally, take the square root of the simplified expression: \( \cos 22.5° = \pm \sqrt{\frac{2 + \sqrt{2}}{4}} \). Choose the positive value since cosine is positive in the first quadrant.

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°).