E. [ Mathematical Communication ] Explain how to transform the graph of f(x)=sin(x) into the graph of g(x)=cos(x). Be specific about distance and direction. Use complete sentences.

Answer 1

If you look at graphs, you will see that we want to move the sin curve π/2 to the left. We do this by having

g(x) = sin(x + π/2)

We can prove that this transformation works by using the sin addition formula

sin(x + π/2) = sin(x)cos(π/2) + cos(x)sin(π/2) = 0 sin(x) + 1 cos(x) = cos(x)

Answer 2

Sin(x) = cos( pi/2 - x)

Sine of the angle is the cosine of the complement