Question

Use a addition or subtraction formula to simply for sin(3theta)*cos(theta)-cos(3theta)*sin(theta)=1/2, then find all solutions within interval [0, 2pi) (PS the simplified part is sin(2theta)=1/2)

Answer 1

3theta and theta and it is cos(3theta-theta) which ends up simplifying to \[\sin(2\theta)=1/2\]

Answer 2

\[
\sin(A-B) = \sin(A)\cos(B) - \cos(A)\sin(B)
\]

Answer 3

I need to find the correct solutions within the interval [0,2pi)

Answer 4

\[
\sin(2\theta) = \frac 12 \\
2\theta = \frac{\pi}{6}, \frac{5\pi}{6}, ....
\]

Answer 5

Add \(2\pi\) to the above to get two more angles.
Then divide them all by 2 to get theta in the interval [0, 2pi]

Answer 6

so the correct answers would be ... pi/12 and 5pi/12 within [0,2pi)

Answer 7

\[\sin(2\theta) = \frac 12 \\
2\theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{\pi}{6} + 2\pi, \frac{5\pi}{6} + 2\pi \\
\theta = \frac{\pi}{12}, \frac{5\pi}{12}, \frac{\pi}{12} + \pi, \frac{5\pi}{12} + \pi \\
\theta = \frac{\pi}{12}, \frac{5\pi}{12}, \frac{13\pi}{12}, \frac{17\pi}{12}
\]

Answer 8

Thanks

Answer 9

You are welcome.