Question about right triangle and sin(2theta)
This is a pretty basic question but I just wanted clarification. I know that sin(theta) is opposite/hypotenuse regarding right triangles. But what would sin(2theta) be? would it be (opposite/hypotenuse) * 2? My problem here is my sin(theta) is x/3. But would sin(2theta) be (2x/3)?
$\endgroup$ 12 Answers
$\begingroup$The number $\sin(2\theta)$ is the sine of twice the angle $\theta$. It is almost never equal to $2\sin(\theta)$.
But there is an important "double-angle" identity $$\sin(2\theta)=2\sin(\theta)\cos(\theta)\tag{1}$$ that you can use in your problem. If you know that $\sin(\theta)=\frac{x}{3}$, all you need to do to find $\sin(2\theta)$ is to find $\cos(\theta)$ and then use the Identity (1).
$\endgroup$ $\begingroup$$\sin 2\theta \not=2\sin \theta$, However 4\sin 2\theta = 2\sin \theta \cos \theta$
As we know, $\cos2\theta \not= 2\cos \theta$, $$\cos2\theta=\sin^2 \theta-\cos^2\theta=2\cos^2\theta-1=1-2\sin^2\theta.$$
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