simple trig equation

**xwrathbringerx** · June 1st 2009, 03:33 AM

sin (x + PI/3) = cos (x – PI/3). Solve for x.

Hi

Could someone please help me solve the above question? For some reason I can't seem to get x.

I end up with:

sin x + root(3) cos x = cos x + root(3) sin x

**Moo** · June 1st 2009, 03:44 AM

Hello,

Originally Posted by xwrathbringerx

sin (x + PI/3) = cos (x – PI/3). Solve for x.

Hi

Could someone please help me solve the above question? For some reason I can't seem to get x.

I end up with:

sin x + root(3) cos x = cos x + root(3) sin x

A way to solve is to note that $\sin(t)=\cos(\pi/2 -t)$

Then $\sin(x+\pi/3)$ can be transformed into $\cos\left(\tfrac\pi 2-\tfrac\pi 3-x\right)=\cos(\pi/6-x)$

Then, two possibilities :
- use the trigonometric identity $\cos(a)-\cos(b)=-2\sin\left(\tfrac{a+b}{2}\right)\sin\left(\tfrac{a-b}{2}\right)$

- or recall that $\cos(a)=\cos(b)\Leftrightarrow a=\pm b+2k\pi\text{ where k is an integer}$ (you can easily see that on a unit circle)

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