Use the given information to find the exact value of each of the following.

Question

a. sin2θ

b. cos2θ

c. tan2θ

cosθ = 5/13, θ lies in Quadrant IV

Answer 1

sinθ = - √1 - 25/169 = - 12/13;

a. sin2θ = 2sinθcosθ = -120/169;

b. cos2θ = 2cos²θ - 1 = 50/169 - 169/169 = - 119/169.

c. tan2θ = sin2θ/cos2θ = 120/119

Answer 2

Hi Richard P.

You should draw a diagram of your angle or triangle in Quadrant IV on a coordinate grid. That said am correct in assuming that you are referring to a Right Triangle in Quadrant IV where x is positive and y is negative? In Quadrant IV both Sin and Tan are negative

If so then the following identities should apply

sin²θ + cos²θ = 1

a. sin2θ = 2sinθcosθ

b. cos2θ = cos²θ - sin²θ

c. tan2θ = (2tanθ)/(1 - tan²θ)

Since you are given cosθ = 5/13 = ADJACENT/HYPOTENUSE for a Right Triangle that means your x coordinate in Quadrant IV is positive 5, length of your adjacent side and your Hypotenuse is 13. You can use the identity above or the Pythagorean Theorem to find the Opposite or or y just keep in mind that in Quadrant IV y is negative. Also if you know a few Pythagorean Triples you would recognize 5, 12, 13 as one of them.

cosθ = 5/13 = 0.38461538

sinθ = -12/13 = -0.923076923

tanθ = (-12/13)/(5/13) = -12/5

Checking in sin²θ + cos²θ = 1

(-12/13)² + (5/13)² = (144/169) + (25/169) = 169/169 = 1

You can plug these into the other identities above

a. sin 2θ = 2(-12/13)(5/13) = -120/169

b. cos 2θ = (25/169) - (144/169) = -119/169

c. tan2θ = (2tanθ)/(1 - tan²θ) can you continue from here?

I hope this helps

If you any questions please contact me.