Answer to Question #74958 in Trigonometry for Anonymous

Question

Simplify: cos⁡(5π+x)


a. cos⁡x b.〖 sin〗⁡x c. -sin⁡x d. -cos⁡x e. None of these

Answer 1

Period of cos(α) is 2π, so we can rewrite cos(5π+x) as
cos(5π+x)=cos(4π+π+x)=cos(π+x)
Then we use the identity
cos(α+β)=cos(α)cos(β)-sin(α)sin(β)
We obtain
cos(π+x)=cos(π)cos(x)-sin(π)sin(x)=(-1)⋅cos(x)-0⋅sin(x)=-cos(x)
Answer: d. cos(5π+x)=-cos(x)