Question

let \\(\theta\\) be the angle in standard position whose terminal side contains the given point. compute \\(\cos(\theta)\\) and \\(\sin(\theta)\\). answer exactly.
(4, -2)
\\(\sin(\theta)=\\)
\\(\cos(\theta)=\\)

Explanation:

Step1: Calculate the radius r

For a point $(x,y)=(4, - 2)$ in the coordinate - plane, the radius $r$ (distance from the origin $(0,0)$ to the point) is given by the formula $r=\sqrt{x^{2}+y^{2}}$.
$r = \sqrt{4^{2}+(-2)^{2}}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}$

Step2: Calculate $\cos(\theta)$

The formula for $\cos(\theta)$ is $\cos(\theta)=\frac{x}{r}$. Substituting $x = 4$ and $r=2\sqrt{5}$, we get $\cos(\theta)=\frac{4}{2\sqrt{5}}=\frac{2}{\sqrt{5}}=\frac{2\sqrt{5}}{5}$

Step3: Calculate $\sin(\theta)$

The formula for $\sin(\theta)$ is $\sin(\theta)=\frac{y}{r}$. Substituting $y=-2$ and $r = 2\sqrt{5}$, we get $\sin(\theta)=\frac{-2}{2\sqrt{5}}=-\frac{1}{\sqrt{5}}=-\frac{\sqrt{5}}{5}$

Answer:

$\sin(\theta)=-\frac{\sqrt{5}}{5}$
$\cos(\theta)=\frac{2\sqrt{5}}{5}$