# How do you find sintheta and costheta if the terminal side of theta lies along the line y=2x in QI?

Apr 29, 2018

The slope, m, of the line $y = 2 x$ is, $m = 2$.

For this line, we know that any ordered pair of coordinates will form a right triangle, therefore, we can choose any point and use the Pythagorean Theorem to compute the hypotenuse:

${h}^{2} = {x}^{2} + {y}^{2}$

Let's choose the point $\left(1 , 2\right)$:

${h}^{2} = {1}^{2} + {2}^{2}$

${h}^{2} = 1 + 4$

${h}^{2} = 5$

$h = \sqrt{5}$

We know that:

$\sin \left(\theta\right) = \frac{y}{h}$

$\sin \left(\theta\right) = \frac{2}{\sqrt{5}}$

Rationalize the denominator:

$\sin \left(\theta\right) = 2 \frac{\sqrt{5}}{5}$

We know that:

$\cos \left(\theta\right) = \frac{x}{h}$

$\cos \left(\theta\right) = \frac{1}{\sqrt{5}}$

Rationalize the denominator:

$\cos \left(\theta\right) = \frac{\sqrt{5}}{5}$