Solve for theta.

Question

6 sin theta-1=2 sin theta +1
0 ≤ theta < 2 pi

Please show the work as equations. Give the exact radian measure and no decimals.

Answer 1

We can solve this equation by isolating the sine function on one side and using trigonometric identities to simplify it:

6 sin theta - 1 = 2 sin theta + 1

4 sin theta = 2

sin theta = 1/2

This means that theta could be in either the first or second quadrant (where sine is positive), and the reference angle for theta is pi/6. Therefore, the solutions are:

theta = pi/6 or theta = 5pi/6

If we add multiples of 2pi to these values, we can get all possible solutions in the given interval of 0 ≤ theta < 2 pi. Therefore, the solutions are:

theta = pi/6 + 2npi or theta = 5pi/6 + 2npi, where n is an integer.