Question

Find all solutions of the equation in radians.
sin(2t)cos(t)-cos(2t)sin(t)=0

Answer 1

1. First simplify the left-hand side of the equation using the trigonometric identity

sin(2t)cos(t) - cos(2t)sin(t) = sin(2t - t) = sin(t)

2. That way the equation becomes sin(t) = 0.

3. The solutions to this equation are t = kπ for all integers k.

4. Therefore, the general solution to the original equation is:

t = kπ or t = π/2 + kπ, where k is an integer.

Answer 2

We can simplify the left-hand side of the equation using the trigonometric identity:

sin(A-B) = sin A cos B - cos A sin B

Using this identity, we can rewrite the left-hand side of the given equation as:

sin(2t - t) = sin(t)

Therefore, we have:

sin(t) = 0

This equation has solutions whenever t is an integer multiple of π, since sin(πk) = 0 for any integer k. Therefore, the solutions of the given equation are:

t = kπ, where k is an integer.