Question

question given the following unit circle, rotate green dot to the appropriate angle and then find the exact value of the function. \\(\cos 240^{\circ}\\)

Explanation:

Step1: Determine the reference angle

The angle $240^\circ$ is in the third quadrant. To find the reference angle, we subtract $180^\circ$ from $240^\circ$:
$240^\circ - 180^\circ = 60^\circ$.

Step2: Recall the cosine value for the reference angle

In the unit circle, $\cos 60^\circ = \frac{1}{2}$.

Step3: Determine the sign of $\cos 240^\circ$

In the third quadrant, both sine and cosine are negative (since $x$ and $y$ coordinates are negative). Thus, $\cos 240^\circ$ will have the same magnitude as $\cos 60^\circ$ but with a negative sign.

So, $\cos 240^\circ = -\cos 60^\circ = -\frac{1}{2}$.

Answer:

$\boldsymbol{-\frac{1}{2}}$