# How to find point of interesection between sin x and sin 0.5x?

• October 15th 2011, 12:57 PM
sorkii
How to find point of interesection between sin x and sin 0.5x?
I'm having trouble manually working out points of intersection between trig graphs. I can do them when its between a cos graph and a sin graph using cos^2(x) + sin^2(x) = 1 but I cant do them when they are the same etc. sin x = sin 2x

How do you manually work these pts of intersection out? Do I have to use the general formula?
• October 15th 2011, 01:25 PM
emakarov
Re: How to find point of interesection between sin x and sin 0.5x?
You can use the fact that sin(2x) - sin(x) = 2sin(x/2)cos(3x/2).
• October 15th 2011, 05:11 PM
Soroban
Re: How to find point of interesection between sin x and sin 0.5x?
Hello, sorkii!

Quote:

$\text{How to find the intersections of }\,y \:=\:\sin x\,\text{ and }\,y \,=\,\sin\tfrac{x}{2}$

We wish to solve: . $\sin x \:=\:\sin\tfrac{x}{2}$

Recall the identity: . $\sin\theta \:=\:2\sin\tfrac{\theta}{2}\cos\tfrac{\theta}{2}$

So we have: . $2\sin\tfrac{x}{2}\cos\tfrac{x}{2} \:=\:\sin\tfrac{x}{2} \quad\Rightarrow\quad 2\sin\tfrac{x}{2}\cos\tfrac{x}{2} - \sin\tfrac{x}{2} \;=\;0$

. . . . . . $\sin\tfrac{x}{2}\big[2\cos\tfrac{x}{2} - 1\big] \;=\;0$

Hence: . $\sin\tfrac{x}{2} \:=\:0 \quad\Rightarrow\quad \tfrac{x}{2} \:=\:\pi n \quad\Rightarrow\quad \boxed{x \:=\:2\pi n}$

And: . $2\cos\tfrac{x}{2}-1 \:=\:0 \quad\Rightarrow\quad \cos\tfrac{x}{2} \:=\:\tfrac{1}{2}$

. . . . . $\tfrac{x}{2} \:=\:\pm\tfrac{\pi}{3}+ 2\pi n \quad\Rightarrow\quad \boxed{x \:=\: \pm\tfrac{2\pi}{3} + 4\pi n}$