Half angle formula

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by schoolmathm


Trig identities are a crucial part of trig functions, and one of the most important identities in trigonometric functions is the half-angle formula. In this post, we discussed various trigonometric identities and formulas. First, we recommend visiting the previous posts to review identities, including the double-angle formula. Here we talk about half-angle formulas. We introduce them, then we demonstrate their use 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 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 $latex\sin (\dfrac{\pi}{12})$ 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 to solve these problems by yourself.

Example 3: If \sin x=\dfrac{2}{3} and $latexx$ 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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