trignometry question

Question

5 cos θ + 12 sin θ = 13, cos θ = ?

Answer 1

I. First method.

5 cos(θ) + 12 sin(θ) = 13 (divide by √(5²+12²)=13 )

(5/13)cos( θ)+(12/13) sin( θ)=1

Denote cos(β)=5/13, then sin(β)=12/13 and

cos(β)cos( θ)+sin(β) sin(θ)=1

cos(β-θ)=1

 β-θ=π/2

 θ=β-π/2

cos(θ)=cos(β-π/2)=cos(β)=5/13

cos(θ)=5/13.

 

II. Another approach

 

5cos(θ)+12sin(θ)=13

5cos(θ)+12√(1-cos²(θ))=13

12²(1-cos²(θ)=(13-5cos(θ))²

12²(1-cos²(θ)=13²-130cos(θ)+25cos²(θ)

13²cos²(θ)-130cos(θ)+25=0

(13cos(θ)-5)²=0

13cos(θ)=5

cos(θ)=5/13.

 

Note: There are many other ways as well.

Answer 2

cosΘ = (-12sinΘ)/5