TI - Half Angle Identities Lesson

Half Angle Identities

The last set of identities we will use are the Half-Angle Identities:

$\sin\left(\frac{\theta}{2}\right)=\pm\sqrt[]{\frac{1-\cos\theta}{2}}\\\cos\left(\frac{\theta}{2}\right)=\pm\sqrt[]{\frac{1+\cos\theta}{2}}$

$\tan\left(\frac{\theta}{2}\right)=\pm\sqrt[]{\frac{1-\cos\theta}{1+\cos\theta}}\\ \tan\left(\frac{\theta}{2}\right)=\frac{1-\cos\theta\:}{\sin\theta}\\ \tan\left(\frac{\theta}{2}\right)=\frac{\sin\theta}{1+\cos\theta}\:$

*To determine which sign to use, you should check the quadrant in which $\frac{\theta}{2}$ lies.

Let's prove $\cos\left(\frac{\theta}{2}\right)=\pm\sqrt[]{\frac{1+\cos\theta}{2}}\:$, we will use our double angle formula for cosine:$\cos\left(2\theta\right)=2\cos^2\theta-1$.

1. In cos(2θ) = 2cos²θ - 1, we know that θ is half of 2θ, so let's set 2θ = x, so that means that θ = x/2. $\cos x=2\cos^2\left(\frac{x}{2}\right)-1$
2. So now let's rearrange this formula to isolate the half angle. $\cos x=2\cos^2\left(\frac{x}{2}\right)-1\\ 1+\cos x=2\cos^2\left(\frac{x}{2}\right)\\ \frac{1+\cos x}{2}=\cos^2\left(\frac{x}{2}\right)\\ \pm\sqrt[]{\frac{1+\cos x}{2}}=\cos\left(\frac{x}{2}\right)$

Watch this video to try using the half-angle formula:

Find the exact value of each expression: 

1. $\sin75°$

• Solution: $\frac{\sqrt[]{2+\sqrt[]{3}}}{2}$

2. $\tan\frac{7\Pi}{12}$

• Solution: $-2-\sqrt[]{3}$

3. $\sin22.5°$

• Solution:  $\frac{\sqrt[]{2-\sqrt[]{2}}}{2}$

Let's try solving an equation using a half-angle identity, watch this video:

Solve each equation on the interval $\left[0,\:2\pi\right)$.

1. Problem: $2\sin^2\frac{x}{2}+\cos x=1+\sin x$

• Solution: $\left\{0,\:\pi\right\}$

2. Problem: $\sin^2\frac{x}{2}=\cos^2\frac{x}{2}$

• Solution: $\left\{\frac{\pi}{2},\:\frac{3\pi}{2}\right\}$

 IMAGES CREATED BY GAVS