ACT Math - How to find positive cosine

To the nearest , what is the cosine formed from the origin to ? Assume counterclockwise rotation.

Explanation

If the point to be reached is , then we may envision a right triangle with sides and , and hypotenuse . The Pythagorean Theorem tells us that , so we plug in and find that:

Thus, $c = \sqrt{61}$

Now, SOHCAHTOA tells us that $\textup{cosine} = \frac{\textup{adjacent}}{\textup{hypotenuse}}$ , so we know that:

$\textup{cos }x^{\circ} = \frac{5}{\sqrt{61}} \approx 0.6402$

Thus, our cosine is approximately .

and is between and $\pi$ . What is the value of ?

$\frac{\sqrt{3}}{2}$

$\sqrt{2}$

$3\sqrt{3}$

Explanation

For to $\pi$ , we know that $cos(\frac{\pi}{3}) = 0.5$ . So, the question asks, what is the value of , where $x=\frac{\pi}{3}$ . Therefore, it is asking what the value of $cos(\frac{\pi}{6})$ is, which is $\frac{\sqrt{3}}{2}$ .

Two drivers race to a finish line. Driver A drives north blocks, and east blocks and crosses the goal. Driver B drives the shortest direct route between the two points. Relative to east, what is the cosine of the angle at which Driver B raced? Round to the nearest .

Explanation

If the point to be reached is blocks north and blocks east, then we may envision a right triangle with sides and , and hypotenuse . The Pythagorean Theorem tells us that , so we plug in and find that: