# sin(theta) is opposite/hypotenuse regarding right triangles. Then what would \sin(2theta)

Ann Tice 2021-11-20 Answered
sin(theta) is opposite/hypotenuse regarding right triangles. Then what would $$\displaystyle{\sin{{\left({2}\theta\right)}}}$$ be?

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Archie Griffin
The number $$\displaystyle{\sin{{2}}}\theta$$ is the sin of twice the angle \theta. It is almost never equal to $$\displaystyle{2}{\sin{\theta}}$$.
$$\displaystyle{\sin{{2}}}\theta={2}{\sin{\theta}}\times{\cos{\theta}}{\left({1}\right)}$$ If you know that $$\displaystyle{\sin{\theta}}={x}{3}$$, all you need to do to find $$\displaystyle{\sin{{2}}}\theta$$ is to find $$\displaystyle{\cos{\theta}}$$.
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Abel Maynard

$$\displaystyle{\sin{{2}}}\theta\ne{2}{\sin{\theta}}$$, but $$\displaystyle{4}{\sin{{2}}}\theta={2}{\sin{\theta}}{\cos{\theta}}$$
As we know, $$\cos2\theta\ne2\cos\theta, \cos2\theta=\sin2\theta−\cos2\theta=2\cos2\theta−1=1−2\sin2\theta$$