If sin(theta)= 2root5 over five and pie over 2 <theta<pie what is cos(theta)?

Question

I have no idea where to even start on this problem.  I know it is asking for cosine(x) and I know my unit circle well, I just don't know what to do.  

Answer 1

sinθ = 2√5/5 and θ lies in quadrant 2.

 

Since θ is a quadrant 2 angle, cosθ < 0.  

 

We know that cos²θ + sin² θ = 1, so cos² θ = 1 - sin² θ

 

So, cosθ = ±√(1 - sin² θ)

 

Since cosθ is negative in quadrant 2, we have cosθ = -√(1 - sin²θ)

 

                                                                       = -√[1 - (2√5/5)²]

 

                                                                       = -√[1 - 4/5]

 

                                                                       = -√(1/5)

 

                                                                       = -√5/5