The Notes: Why does y=(sinx)^2+(cosx)^2 create a flat line on a graph?

Saturday, May 23, 2009

Why does y=(sinx)^2+(cosx)^2 create a flat line on a graph?

Hello!

A flat line (a straight horizontal line) as a graph of $\displaystyle{y}={f{{\left({x}\right)}}}$ means that $\displaystyle{f{{\left({x}\right)}}}$ is a constant (all its values are the same). Is this true for $\displaystyle{y}={{\sin}}^{{2}}{\left({x}\right)}+{{\cos}}^{{2}}{\left({x}\right)}$ ? Yes, it is one of the most known trigonometric identities.

It is simple to prove it, at least for $\displaystyle{x}$ between $\displaystyle{0}$ and $\displaystyle\frac{\pi}{{2}}$ (90 degrees). Such an $\displaystyle{x}$ may be an angle in a right triangle. If the lengths of its legs are $\displaystyle{a}$ and $\displaystyle{b},$ then the length of the hypotenuse is $\displaystyle\sqrt{{{{a}}^{{2}}+{{b}}^{{2}}}}$ by the Pythagorean theorem.

Let $\displaystyle{a}$ be the adjacent leg of the angle $\displaystyle{x},$ then

$\displaystyle{\cos{{\left({x}\right)}}}=\frac{{a}}{\sqrt{{{{a}}^{{2}}+{{b}}^{{2}}}}}$ and $\displaystyle{\sin{{\left({x}\right)}}}=\frac{{b}}{\sqrt{{{{a}}^{{2}}+{{b}}^{{2}}}}}.$

Therefore $\displaystyle{{\cos}}^{{2}}{\left({x}\right)}=\frac{{{a}}^{{2}}}{{{{a}}^{{2}}+{{b}}^{{2}}}}$ and $\displaystyle{{\sin}}^{{2}}{\left({x}\right)}=\frac{{{b}}^{{2}}}{{{{a}}^{{2}}+{{b}}^{{2}}}},$