In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression. sin 22.5°

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

Set \(\theta = 45^\circ\) in the half-angle formula, so \(\sin 22.5^\circ = \sin\left(\frac{45^\circ}{2}\right) = \pm \sqrt{\frac{1 - \cos 45^\circ}{2}}\).

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 to get \(\sin 22.5^\circ = \pm \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}\).

Determine the correct sign of the square root based on the quadrant of 22.5° (which is positive in the first quadrant), so take the positive root.

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Half-Angle Formulas

Half-angle formulas allow you to find the sine, cosine, or tangent of half an angle using the trigonometric values of the original angle. For sine, the formula is sin(θ/2) = ±√((1 - cos θ)/2), where the sign depends on the quadrant of θ/2.

Exact Values of Common Angles

Knowing the exact trigonometric values of common angles like 45°, 30°, and 60° is essential. For example, cos 45° = √2/2, which is used in half-angle formulas to find values like sin 22.5° (half of 45°).

Sign Determination in Trigonometry

When using half-angle formulas, determining the correct sign (positive or negative) is crucial. This depends on the quadrant where the resulting angle lies; since 22.5° is in the first quadrant, sine is positive.