Finding sin(15°)

Find the exact value of $\sin 15°$.

Express as half angle: $15° = \frac{30°}{2}$, so $\theta = 30°$ = $\sin 15° = \sin\frac{30°}{2}$

Find cos(30°): From the unit circle: $\cos 30° = \frac{\sqrt{3}}{2}$ = $\cos\theta = \frac{\sqrt{3}}{2}$

Apply half-angle formula: $\sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}} = \pm\sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}}$ = $= \pm\sqrt{\frac{\frac{2-\sqrt{3}}{2}}{2}}$

Simplify: $= \pm\sqrt{\frac{2-\sqrt{3}}{4}} = \pm\frac{\sqrt{2-\sqrt{3}}}{2}$ = $= \frac{\sqrt{2-\sqrt{3}}}{2}$

Determine the sign: $15°$ is in Quadrant I, where sine is positive = Use $+$

Answer: $\sin 15° = \frac{\sqrt{2-\sqrt{3}}}{2} \approx 0.2588$

Finding cos(22.5°)

Find the exact value of $\cos 22.5°$.

Express as half angle: $22.5° = \frac{45°}{2}$, so $\theta = 45°$ = $\cos 22.5° = \cos\frac{45°}{2}$

Find cos(45°): From the unit circle: $\cos 45° = \frac{\sqrt{2}}{2}$ = $\cos\theta = \frac{\sqrt{2}}{2}$

Apply half-angle formula: $\cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}} = \pm\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$ = $= \pm\sqrt{\frac{\frac{2+\sqrt{2}}{2}}{2}}$

Simplify: $= \pm\sqrt{\frac{2+\sqrt{2}}{4}} = \pm\frac{\sqrt{2+\sqrt{2}}}{2}$ = $= \frac{\sqrt{2+\sqrt{2}}}{2}$

Determine the sign: $22.5°$ is in Quadrant I, where cosine is positive = Use $+$

Answer: $\cos 22.5° = \frac{\sqrt{2+\sqrt{2}}}{2} \approx 0.9239$

Finding tan(π/8)

Find the exact value of $\tan\frac{\pi}{8}$.

Express as half angle: $\frac{\pi}{8} = \frac{\pi/4}{2}$, so $\theta = \frac{\pi}{4}$ = $\tan\frac{\pi}{8} = \tan\frac{\theta}{2}$

Find sin and cos of θ: $\sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}$, $\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$ = Both equal $\frac{\sqrt{2}}{2}$

Apply tan half-angle formula: $\tan\frac{\theta}{2} = \frac{1 - \cos\theta}{\sin\theta} = \frac{1 - \frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}$ = $= \frac{\frac{2-\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}$

Simplify: $= \frac{2-\sqrt{2}}{\sqrt{2}} = \frac{(2-\sqrt{2})\sqrt{2}}{2} = \frac{2\sqrt{2}-2}{2}$ = $= \sqrt{2} - 1$

Answer: $\tan\frac{\pi}{8} = \sqrt{2} - 1 \approx 0.4142$

Mistake: Forgetting to determine the correct sign

Why: The half-angle formulas give $\pm$, and students often just use positive. The sign depends on which quadrant $\frac{\theta}{2}$ is in, not $\theta$.

Correct: Always identify the quadrant of the half angle first, then apply the appropriate sign for that quadrant.

Mistake: Using the wrong formula for tangent

Why: There are two forms of the tangent half-angle formula. Some students try to derive it from sin/cos half-angles, which is more complex.

Correct: Use $\tan\frac{\theta}{2} = \frac{1 - \cos\theta}{\sin\theta}$ or $\frac{\sin\theta}{1 + \cos\theta}$ directly.

Mistake: Confusing half-angle with double-angle formulas

Why: Half-angle formulas involve square roots; double-angle formulas do not. Students sometimes mix them up.

Correct: Half-angle: has $\sqrt{}$ and $\pm$. Double-angle: no square root, uses $2\theta$.

Signal Processing

In electronics and telecommunications, half-angle identities help analyze and synthesize waveforms.

When combining two radio signals, engineers use half-angle formulas to predict interference patterns and optimize signal strength.

Architecture and Design

Architects use half-angle calculations when designing structures with specific angular measurements.

A geodesic dome requires precise angle calculations. If a structural element meets at $45°$, the half-angle of $22.5°$ determines the cut angle for supporting beams.

Q: Why do half-angle formulas have a ± sign but double-angle formulas don't?

A: Half-angle formulas involve taking a square root, which always produces a positive value. The ± accounts for the fact that the actual trig value could be negative (depending on quadrant). Double-angle formulas don't have square roots, so no ambiguity arises.

Q: How do I know which tangent half-angle formula to use?

A: Both forms give the same answer. Use $\frac{1-\cos\theta}{\sin\theta}$ when you want to avoid division by a small number (when $\cos\theta \approx -1$), and use $\frac{\sin\theta}{1+\cos\theta}$ when $\cos\theta \approx 1$.

Q: Can I use half-angle formulas for any angle?

A: Yes! As long as you know the trig values for the original angle $\theta$, you can find the values for $\frac{\theta}{2}$. This is especially useful for angles not on the standard unit circle.