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 (Headbang)

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- Jun 1st 2009, 03:33 AMxwrathbringerxsimple trig equation
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 (Headbang) - Jun 1st 2009, 03:44 AMMoo
Hello,

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

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

Then, two possibilities :

- use the trigonometric identity $\displaystyle \cos(a)-\cos(b)=-2\sin\left(\tfrac{a+b}{2}\right)\sin\left(\tfrac{a-b}{2}\right)$

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