can someone help me with trigonometric equations?

Question

Solve each equation on the interval 0≤θ<2pi

tan θ=2 sin θ

Answer 1

tan x = 2 sin x          I am going to use x since it is quicker for me to type.

tan x - 2 sin x = 0        move everything over to one side

(sin x)/(cos x) - 2 sin x = 0          write tangent in terms of sines and cosines

sin x (1/cos x  - 2) = 0    factor out sine

sin x = 0 or 1/cos x - 2 = 0       set each factor equal to zero

sin x = 0   what angles give you a sine value of zero? x = n * pi  where n is an integer.

We are looking for angles in [0, 2pi) so 0*pi and 1*pi are the only angles in that interval. So, 0 and pi are two of our solutions.

1/cos x - 2 = 0    cos x = 1/2     your reference angle is pi/3  

cosine is positive in quadrants 1 and 4, so pi/3 and 5*pi/3 are the angles in our interval.

x = 0, pi, pi/3 and 5*pi/3 are the solutions.