Writing expressions in terms of only sine

Question

If I were to do this without these formulas, I would pull out a number that made both of the numbers(like (sqrt(3))/2 and 1/2) in the picture would be something that I could get a sine and cosine that works for them. This formula for K gives me this number I should pull out but how? Can someone explain why the formula for K is the number that you should pull out to make the numbers work for a cosine and sine insertion?

[IMG]http://i57.tinypic.com/2n1gwty.jpg[/IMG]

Answer 1

You want to adjust the coefficients so that the sum of their squares is $1$. That makes them the sine and cosine of some angle. So multiply and divide by the square root of the sum of their squares: $$a\sin t + b\cos t=\sqrt{a^2+b^2}\left(\frac{a}{\sqrt{a^2+b^2}}\sin t +\frac{b}{\sqrt{a^2+b^2}}\cos t\right) $$ $$=\sqrt{a^2+b^2}\left(\sin t\cos \phi +\cos t\sin \phi\right)=\sqrt{a^2+b^2}\sin(t+\phi)$$ where $\phi$ is the angle with $(\cos\phi,\sin\phi)=(a/\sqrt{a^2+b^2},b/\sqrt{a^2+b^2})$. This is possible because $$\left(\frac{a}{\sqrt{a^2+b^2}}\right)^2+\left(\frac{b}{\sqrt{a^2+b^2}}\right)^2 =\frac{a^2+b^2}{a^2+b^2}=1$$ so the adjusted coefficients are the coordinates of a point on the unit circle.