Question

Use a double- or half-angle formula to solve the equation in the interval [0, 2π). sin 3θ -sin 6θ = 0

Answer 1

To solve the equation sin 3\u03b8 - sin 6\u03b8 = 0, use the double-angle formula to express sin 6\u03b8 and then factor and solve the resulting equations for \u03b8 in the interval [0, 2\u03c0).

Step-by-step explanation:

To solve the equation sin 3\u03b8 - sin 6\u03b8 = 0 using a double-angle formula, we can approach it step-by-step:

1. 
   

First, recognize that sin 6\u03b8 can be expressed in terms of sin 3\u03b8 using the double-angle formula for sine, which is sin 2\u03b1 = 2sin \u03b1 cos \u03b1. Thus, sin 6\u03b8 = 2sin 3\u03b8 cos 3\u03b8.

Solve this equation by setting each factor equal to 0, resulting in sin 3\u03b8 = 0 and 1 - 2cos 3\u03b8 = 0.

To solve 1 - 2cos 3\u03b8 = 0, we get cos 3\u03b8 = 1/2, leading to \u03b8 = \u03c0/9, 5\u03c0/9 when considering the interval [0, 2\u03c0).

The final step is to list all the solutions which are \u03b8 = 0, \u03c0/3, 2\u03c0/3, \u03c0/9, 5\u03c0/9, within the interval [0, 2\u03c0).