Find the exact values of sin(2θ), cos(2θ), and tan(2θ) given sec(θ) = − 13/5 and θ is in the interval [π 2 , π ]

Question

My professor didn't explain this question well in class so I would appreciate if someone could show me how to do it step-by-step. Thank you!

Answer 1

You need to use the double angle formulas on this.

First, given sec(θ) = − 13/5 in Q2 (between π/2 and π) we have this (since sec = hyp/adj)

Using the Pythagorean Theorem, we can solve for the "opp" and "opp" = 12.

So, cos(θ) = -5/13, sin(θ) = 12/13, tan(θ) = -12/5

Now, to calculate sin(2θ) we use sin(2θ) = 2sin(θ)cos(θ) = 2(12/13)(-5/13) = -120/169

To calculate cos(2θ) we can use one of several but I'll pick cos(2θ) = cos²(θ) – sin²(θ) = (-5/13)² - (12/13)² = 25/169 - 144/169 = -119/169

To calculate tan(2θ) we use tan(2θ) = (2tan(θ))/(1-tan²(θ)) = ((2)(-12/5))/(1-(-12/5)²) = (-24/5)/(-119/25) = 120/119