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
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)