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

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Recognize that 22.5° is half of 45°, so you can use the half-angle formula for cosine: \(\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos\theta}{2}}\).

Identify \(\theta = 45^\circ\), so the expression becomes \(\cos 22.5^\circ = \pm \sqrt{\frac{1 + \cos 45^\circ}{2}}\).

Recall the exact value of \(\cos 45^\circ\), which is \(\frac{\sqrt{2}}{2}\).

Substitute \(\cos 45^\circ = \frac{\sqrt{2}}{2}\) into the formula: \(\cos 22.5^\circ = \pm \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}\).

Determine the correct sign for the square root based on the quadrant where 22.5° lies (first quadrant, where cosine is positive), so choose the positive root.

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Half-Angle Formulas

Half-angle formulas express trigonometric functions of half an angle in terms of the function of the original angle. For cosine, the formula is cos(θ/2) = ±√[(1 + cos θ)/2]. The sign depends on the quadrant of the half-angle. These formulas help find exact values for angles like 22.5°, which is half of 45°.

Exact Values of Common Angles

Knowing the exact trigonometric values of standard angles such as 0°, 30°, 45°, 60°, and 90° is essential. For example, cos 45° = √2/2. These values serve as the basis for applying half-angle formulas to find exact values of non-standard angles like 22.5°.

Sign Determination in Trigonometric Functions

When using half-angle formulas, determining the correct sign (positive or negative) is crucial. This depends on the quadrant where the half-angle lies. Since 22.5° is in the first quadrant, where cosine is positive, the positive root is chosen for cos 22.5°.