Mastering Half-Angle Identities: Pre-Calculus Explanations and Tips

📚 Understanding Half-Angle Identities

Half-angle identities are trigonometric formulas that relate the trigonometric functions of an angle to those of half of that angle. They are derived from the double-angle formulas and are useful when you need to find the trigonometric function of an angle that is half of a known angle. Let's dive in!

🧮 The Formulas

• ➕ Sine Half-Angle: $\sin(\frac{x}{2}) = \pm \sqrt{\frac{1 - \cos(x)}{2}}$
• ➖ Cosine Half-Angle: $\cos(\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos(x)}{2}}$
• ➗ Tangent Half-Angle: $\tan(\frac{x}{2}) = \frac{\sin(x)}{1 + \cos(x)} = \frac{1 - \cos(x)}{\sin(x)}$

The $\pm$ sign indicates that you need to determine the correct sign based on the quadrant in which $\frac{x}{2}$ lies.

📝 Tips for Remembering

• 💡 Sine is Negative: Think of sine as being related to subtraction in the numerator (1 - cos(x)).
• ➕ Cosine is Positive: Cosine is related to addition in the numerator (1 + cos(x)).
• 🧭 Quadrant Matters: Always check the quadrant of $\frac{x}{2}$ to determine the correct sign (+ or -) for sine and cosine.
• 🤝 Tangent Variations: Remember that the tangent half-angle identity has two common forms; choose the one that best suits the problem.

➗ Examples

Let's find $\sin(15^\circ)$ using the half-angle identity. Since $15^\circ = \frac{30^\circ}{2}$, we can use the half-angle formula for sine:

$\sin(15^\circ) = \sin(\frac{30^\circ}{2}) = \sqrt{\frac{1 - \cos(30^\circ)}{2}}$

We know that $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, so:

$\sin(15^\circ) = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{3}}{4}} = \frac{\sqrt{2 - \sqrt{3}}}{2}$

Since $15^\circ$ is in the first quadrant, the sine is positive.

✍️ Practice Quiz

Evaluate the following using half-angle identities:

1. ❓ Find $\cos(15^\circ)$.
2. ❓ Find $\tan(15^\circ)$.
3. ❓ Find $\sin(22.5^\circ)$.
4. ❓ Find $\cos(22.5^\circ)$.
5. ❓ Find $\tan(22.5^\circ)$.
6. ❓ Find $\sin(\frac{\pi}{8})$.
7. ❓ Find $\cos(\frac{\pi}{8})$.

🔑 Solutions

1. $\cos(15^\circ) = \frac{\sqrt{2 + \sqrt{3}}}{2}$
2. $\tan(15^\circ) = 2 - \sqrt{3}$
3. $\sin(22.5^\circ) = \frac{\sqrt{2 - \sqrt{2}}}{2}$
4. $\cos(22.5^\circ) = \frac{\sqrt{2 + \sqrt{2}}}{2}$
5. $\tan(22.5^\circ) = \sqrt{2} - 1$
6. $\sin(\frac{\pi}{8}) = \frac{\sqrt{2 - \sqrt{2}}}{2}$
7. $\cos(\frac{\pi}{8}) = \frac{\sqrt{2 + \sqrt{2}}}{2}$