SOLUTION: Use an identity to evaluate sin2θ when the angle, θ terminates in quadrant III and tanθ=4/3. (I am sure I need to use the identity sin(2θ) = 2 sinθ cosθ, but I am unsure how

Algebra ->  Trigonometry-basics -> SOLUTION: Use an identity to evaluate sin2θ when the angle, θ terminates in quadrant III and tanθ=4/3. (I am sure I need to use the identity sin(2θ) = 2 sinθ cosθ, but I am unsure how       Log On


   

Question 1174752: Use an identity to evaluate sin2θ when the angle, θ terminates in quadrant III and tanθ=4/3. (I am sure I need to use the identity sin(2θ) = 2 sinθ cosθ, but I am unsure how use this for tan)
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
you are given that tan(theta) = 4/3.

to find the angle, solve for arctan(4/3) using your calculator.
you will find that arctan(4/3) = 53.13010235 degrees.

that's the angle in the first quadrant.
the equivalent angle in the third quadrant is 180 + 53.13010235 = 233.13010235 degrees.

you are now looking for sin(2 * theta) when theta = 233.13010235 degrees.
2 * theta = 466.2602047 degrees.
you get sin(2 * 233.13010235) = sin(466.2602047)) = .96.
that should be your answer.

if you used the sin(2 * theta) identity, you would have gotten:
sin(2 * theta) = 2 * sin(theta) * cos(theta) which becomes:
sin(2 * theta) = 2 * sin(233.13010235) * cos(233.13010235) = .96.

that's the same as we got before, so it should be good.