solve the following system of equations algebraically (-2π, 2π)
y=cos^2 (x)
y=sin^2 (x) + 1/2
please show all work
Thanks
Hello, meli3000!
$\displaystyle \text{Add {\color{blue}[1]} and {\color{blue}[2]}: }\;2y \;=\;\underbrace{\cos^2\!x+\sin^2\!x}_{\text{This is 1}} + \frac{1}{2} \quad\Rightarrow\quad 2y \:=\:\frac{3}{2} \quad\Rightarrow\quad\boxed{y \:=\:\frac{3}{4}}$Solve the following system of equations algebraically $\displaystyle (-2\pi,\:2\pi)$
. . $\displaystyle {\color{blue}[1]}\;\;y \;=\;\cos^2\!x$
. . $\displaystyle {\color{blue}[2]}\;\;y\:=\:\sin^2\!x + \frac{1}{2}$
Substitute into [1]: .$\displaystyle \cos^2\!x \:=\:\frac{3}{4}\quad\Rightarrow\quad \cos x \:=\:\pm \frac{\sqrt{3}}{2}$
. . $\displaystyle \boxed{x \;=\;\pm\frac{\pi}{6},\:\pm\frac{5\pi}{6},\:\pm\fr ac{7\pi}{6},\:\pm\frac{11\pi}{6}}$