Explain why sin?^2 + cos?^2=1

Question

Explain why sin⊗² + cos⊗²=1

Answer 1

Let θ be an angle in standard position with vertex at (0,0).  Let (x, y) be a point on the terminal side of θ and let r be the distance from (0,0) to (x,y).

 

By the distance formula, the distance from (x,y) to (0,0) is                                       r = √[(x-0)² + (y-0)²] = √[x²+y²].  So, x² + y² = r²

 

By definition, cosθ = x/r and sinθ = y/r

 

So, cos²θ + sin² θ = (x/r)² + (y/r)² = (x²+y²)/r² = r²/r² = 1

 

  

Answer 2

Imagine a right triangle with sides x, y, and √(x² + y²). (The third side comes from the pythagorean theorem:

a² + b² = c²).

 

Then, for a given angle in the triangle θ:

 

                sin (θ) = x/√(x² + y²)

 

                cos (θ) = y/√(x² + y²)

 

So,

 

               sin²(θ) + cos²(θ) =

 

              x²/(x² + y²) + y²/(x² + y²) =

 

              (x² + y²)/(x² + y²) =

 

              1. 

 

-Ray Biggerstaff