Question

A line through the origin and (10,4) is showin in the standard (x,y) coordinate plane below. The acute angle between the line and the positive x-axis has measure θ.

Answer 1

Answer:

$\cos\theta=\frac{10}{\sqrt{116}}$

Step-by-step explanation:

To solve this, we can imagine a line drawn directly down from the point (10,4) to the x-axis. This would make a right triangle with length 10 and height 4.

$\cos\theta$ is equal to the length of adjacent leg to the angle divided by the length of the hypotenuse of the triangle. From the picture, we can see the adjacent leg is 10.

We can use Pythagorean's Theorem to find the hypotenuse (let $c$ represent the length of the hypotenuse):

$10^2+4^2=c^2$

$100+16=c^2$

$116=c^2$

$c=\sqrt{116}$

Thus, we know the adjacent is 10, and the hypotenuse is $\sqrt{116}$. This means $\cos\theta=\frac{10}{\sqrt{116}}$.

Hope this helps!