Find the exact value of each. cos 𝜽 = -5/13 where 𝛑/2 ≤ 𝜽 < 𝛑 Find tan 2𝜽

Question

I would appreciate any help with this. Preferably, showing the work so I can better understand it.

Answer 1

cosx = -5/13 in the 3rd quadrant

It's a 5-12 -13 right triangle, 5^2 + 12^2 = 13^2, 25+144 = 169

sinx = 12/13

sin2x = 2sinxcosx = 2(-5/13)(12/13) = -120/169

cos2x = cos^2x - sin^2x = (-5/13)^2 - (12/13)^2 = 25/169-144/169 = -119/169

tan2x = sin2x/cos2x = -120/169 over -119/169 = 120/119

tan(2theta) = 120/119

just to check the answer

find theta using an inverse trig function on a scientific calculator: sinx = 12/13 x = about 67.38 degrees, but you want x in the 3rd quadrant so x = 180-67.38 =

2x = 225.24 degrees

tan2x = 1.008

120/119 = 1.008

exact answer is tan(2theta) = 120/119