Math Snap

STEP 1

1. The angle $300^\circ$ is measured in standard position.
2. The cosecant function, $\csc \theta$, is the reciprocal of the sine function, $\sin \theta$.

STEP 2

1. Determine the reference angle for $300^\circ$.
2. Identify the quadrant in which $300^\circ$ lies.
3. Calculate $\sin 300^\circ$ using the reference angle and quadrant information.
4. Find $\csc 300^\circ$ by taking the reciprocal of $\sin 300^\circ$.
5. Choose the correct answer from the given options.

STEP 3

Determine the reference angle for $300^\circ$.
The reference angle is calculated as:
$$360^\circ - 300^\circ = 60^\circ$$

STEP 4

Identify the quadrant in which $300^\circ$ lies.
Since $300^\circ$ is between $270^\circ$ and $360^\circ$, it lies in the fourth quadrant.

STEP 5

Calculate $\sin 300^\circ$ using the reference angle and quadrant information.
In the fourth quadrant, the sine function is negative. Therefore:
$$\sin 300^\circ = -\sin 60^\circ = -\frac{\sqrt{3}}{2}$$

STEP 6

Find $\csc 300^\circ$ by taking the reciprocal of $\sin 300^\circ$.
$$\csc 300^\circ = \frac{1}{\sin 300^\circ} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}}$$

Choose the correct answer from the given options.
The correct answer is:
c) $-\frac{2 \sqrt{3}}{3}$