Half angle formula

Trig identities are crucial part of trig functions and one of the most important identities in trigonometric functions is the half angle formula. In this post, we talked about different trig identities and formulas. First we recommend to visit the previous posts to review identities, including the double angle formula. Here we talk about half angle formulas. We introduce them, then we see how to use them with some examples and applications.

What is the half angle formula?

There are 3 main half angle formulas for sin, cos and tan:

  • $\sin^{2} u=\frac{1-\cos (2 u)}{2}$
  • $\cos^{2} u=\frac{1+\cos (2 u)}{2}$
  • $\tan^{2} u=\frac{1-\cos (2 u)}{1+\cos (2 u)}$

Next we see some good and useful examples of half angle formula. First, try to solve them by yourself and then check the solutions carefully.

Examples of half angle formula:

Example 1: Find $\sin (\dfrac{\pi}{12})$.

Solution: To solve this problem, we use the half angle formula for the function $\sin$.
$$\sin^{2}(\dfrac{\pi}{12})=\frac{1-\cos (2 \dfrac{\pi}{12})}{2}=\frac{1-\cos (\dfrac{\pi}{6})}{2}$$
We know that $\cos (\dfrac{\pi}{6})=\dfrac{\sqrt{3}}{2}$. So $$\sin^{2}(\dfrac{\pi}{12})=\dfrac{1-\dfrac{\sqrt{3}}{2}}{2}=\dfrac{2-\sqrt{3}}{4}$$
Since $\pi/12$ is in the first quadrant, then $\sin (\dfrac{\pi}{12})$ is positive, so
$$\sin^{2}(\dfrac{\pi}{12})=\sqrt{ \dfrac{2-\sqrt{3}}{4} }$$

Example 2: Prove the identity $2\cos^2 x \sec(2x)=\sec(2x) +1$.

Proof: We start from the left side and then we prove the right side. For the $\cos^2 x$ we use half angle formula, so $$2\cos^2 x \sec(2x)=(1+\cos (2 x))\sec(2x)=\sec(2x)+\sec(2x)\cos (2 x)=\sec(2x) +1$$

We add more examples of the half angle formula for you to practice. To learn about this topic it is really important to try solve these problems by yourself.

Example 3: If $\sin x=\dfrac{2}{3}$ and $x$ is in the first quadrant, find $\sin(\frac{x}{2})$.

Example 4: Find the exact value of $\tan(\dfrac{\pi}{8})$.

Example 5: Find a half angle formula for $\cot x$.

If you need help to understand the trig identities, double angle formulas and half angle formulas, contact us.

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