Question

Write an expression that gives all solutions to the equation.
cos x - sin x = 2^(1/2)

Answer 1

I'm having trouble figuring out how to write this problem out. I'm given five different answers to select.
a. pi/4 + 2kpi
b. 7pi/4 + kpi
c. pi/4 + kpi
d. 7pi/4 + 2kpi
e. 3pi/4 + 2kpi

Answer 2

let:\[y=\cos(x)\]which means we have:\[\sin(x)=\sqrt{1-\cos^2(x)}=\sqrt{1-y^2}\]using these substitutions your equation becomes:\[y-\sqrt{1-y^2}=\sqrt{2}\]re-arranging we get:\[\sqrt{1-y^2}=y-\sqrt{2}\]then square both sides:\[1-y^2=(y-\sqrt{2})^2=y^2-2y\sqrt{2}+2\]which leads to:\[2y^2-2y\sqrt{2}+1=0\]solve this quadratic to find y.
then find x.