If csc theta=13/5 and cos theta>0 what is sin 2theta

Answer 1

csc(θ) = 1/sin(θ) and sin(θ) = opposite/hypotenuse (or O/H) so that means csc(θ) = H/O so since csc(θ) = 13/5 then that means the hypotenuse = 13 and the opposite = 5.

So the triangle looks like this:

To determine sin(2θ), you will need to use the double angle equations:

sin(2θ) = 2sin(θ)cos(θ) in the triangle above, we can calculate the other side of the triangle using the Pythagorean Theorem and it turns out to be 12. So the sin(θ) = 5/13 and the cos(θ) = 12/13. So sin(2θ) = (2)(5/13)(12/13) = 120/169

Answer 2

csc θ = 13/5

sin θ = 5/13

cos θ = 12/13

sin 2θ = 2 sin θ cos θ Identity

sin 2θ = 2(5/13)(12/13)