If sec 𝜃 =2 and csc𝜃 <0, find (a) sin(2𝜃) (b) cos(2𝜃) (c) tan(2𝜃)

Question

can you do step by step please?

Answer 1

Lola,

if you don’t recognize a friendly angle, you need to know double angle formula.

But in this case, you know the ratios in a 30-60-90 triangle.

First let’s find the acute angle: sec(theta) = 2 means that cos(theta) = 1/2, which means theta = 60 degrees

Which quadrant has positive sec (or cos) and negative csc ( or sin)? quadrant IV.

so theta = 300 degrees and 2* theta = 600 degrees which has same ratios as 240 degrees.

sin(240) = - sin(60) = - sqrt(3)/2

cos(240) = - cos(60) = - 1/2

tan(240) = sin(240) / cos(240) = sqrt(3)

Answer 2

secθ=2 same as cosθ=1/2

cscθ<0 same as sinθ-<0

angle is -pi/3 or 5pi/3 (same angle termination)

2θ is -2pi/3 or 10pi/3 which are both the same termination as 4pi/3)

one way to solve is to just write down cos(4pi/3)=-1/2,,,sin(4pi/3)=-(√3)/2,,,tan(4pi/3)=√3

or you can use formulas

cos2θ=cos²θ-sin²θ

here cos(10pi/3)=(1/2)²-(-(√3)/2)²=-1/2

yay, same answer

given sin2θ=2sinθcosθ

should find same answer for sin(10pi/3) where 10pi/3 is same as 4pi/3 and -2pi/3

finally tanθ=-√3 since cosθ=1/2 and sjinθ=-(√3)/2

tan2θ=2tanθ/(1+tan²θ)