# How many common points have f and g function graphs in the interval <0;2pi> ?

## a) $f \left(x\right) = \cos x$ $g \left(x\right) = 1 - \sin x$ I can do this by graphing, but I want to know how to do this without graphs.

Oct 22, 2017

intersection points are $\left(0 , 1\right)$ and $\left(\frac{\pi}{2} , 0\right)$

#### Explanation:

We can solve algebraically. We know that $y = \cos x$ and $y = 1 - \sin x$, so we can substitute $f \left(x\right)$ into $g \left(x\right)$ or vice versa.

$\cos x = 1 - \sin x$

$\cos x + \sin x = 1$

${\cos}^{2} x + {\sin}^{2} x + 2 \sin x \cos x = 1$

$1 + 2 \sin x \cos x = 1$

$2 \sin x \cos x = 0$

$\sin \left(2 x\right) = 0$

$2 x = 0 \mathmr{and} \pi$

$x = 0 \mathmr{and} \frac{\pi}{2}$

We check to make sure neither roots are extraneous:

cos(0) + sin(0) =^? 1

1 + 0 = 1 color(green)(√)

ALSO

cos(pi/2) + sin(pi/2) =^? 1

0 + 1 = 1 color(green)(√)

Hence, the graphs intersect at $\left(0 , 1\right)$ and $\left(\frac{\pi}{2} , 0\right)$. The graph confirms. Hopefully this helps!