Half-Angle Identities

To find the trigonometric ratios of half of the standard angles, we use half-angle formulas. In this step-by-step guide, you will learn more about the half-angle formulas.

Step-by-step guide tohalf-angle identities

The half angle formulas are derived from double angle formulas and are expressed in terms of half angles such as $$\frac{\theta }{2}, \frac{x}{2}, \frac{A}{2}$$. Half-angle formulas are used to find the exact values of trigonometric ratios of angles such as $$22.5°$$ (which is half of the standard angle of $$45°$$), $$15°$$ (which is half of the standard angle of $$30°$$).

Here is a list of important half-angle formulas:

• $$\color{blue}{sin\:\left(\frac{A}{2}\right)=\pm \sqrt{\frac{1-\:cos\:A}{2}}}$$
• $$\color{blue}{cos\:\left(\frac{A}{2}\right)=\pm \sqrt{\frac{1+\:cos\:A}{2}}}$$
• $$\color{blue}{tan\:\left(\frac{A}{2}\right)=\pm \sqrt{\frac{1-\:cos\:A}{1+cos\:A}}=\frac{sin\:A}{1+cos\:A}=\frac{1-cos\:A}{sin\:A}}$$

Half-Angle Identities– Example 1:

Find $$sin\:15^{\circ }$$ using a half-angle formula.

Solution:

Since $$15^{\circ }=\frac{30^{\circ }}{2}$$, we can use this formula: $$sin\:\left(\frac{A}{2}\right)=\pm \sqrt{\frac{1-\:cos\:A}{2}}$$

$$sin \frac{30^{\circ }}{2}=\sqrt{\frac{1-\:cos\:30^{\circ }}{2}}$$

$$=\sqrt{\frac{1-\sqrt{\frac{3}{2}}}{2}}$$

$$=\sqrt{\frac{\frac{2-\sqrt{3}}{2}}{2}}$$

$$=\sqrt{\frac{2-\sqrt{3}}{4}}$$

$$=\frac{\sqrt{2-\sqrt{3}}}{2}$$

we used the positive root because $$sin\:15^{\circ }$$ is positive.

Exercises forHalf-Angle Identities

Find each value using a half-angle formula.

1. $$\color{blue}{sin\:165^{\circ }}$$
2. $$\color{blue}{tan\:\frac{\pi }{8}}$$
3. $$\color{blue}{sin\:67.5^{\circ }}$$
4. $$\color{blue}{tan\:\frac{7\pi }{8}}$$
1. $$\color{blue}{\frac{\sqrt{6}-\sqrt{2}}{4}}$$
2. $$\color{blue}{\sqrt{2}-1}$$
3. $$\color{blue}{\frac{\sqrt{2+\sqrt{2}}}{2}}$$
4. $$\color{blue}{1-\sqrt{2}}$$

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