Question

find the exact values of the six trigonometric functions of the given angle. do not use a calculator. 300°. select the correct choice below and fill in any answer boxes within your choice. csc 300° = -\frac{2\sqrt{3}}{3} (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) a. the function value is undefined. select the correct choice below and fill in any answer boxes within your choice. sec 300° = 2 (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) a. the function value is undefined. select the correct choice below and fill in any answer boxes within your choice. cot 300° = (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) a. the function value is undefined.

Explanation:

Step1: Determine the reference angle

The angle $300^{\circ}$ is in the fourth - quadrant. The reference angle $\theta_{r}=360^{\circ}-300^{\circ} = 60^{\circ}$.

Step2: Recall the definitions of trigonometric functions

• $\csc\theta=\frac{1}{\sin\theta}$, $\sec\theta=\frac{1}{\cos\theta}$, $\cot\theta=\frac{\cos\theta}{\sin\theta}$.
• In the fourth - quadrant, $\sin300^{\circ}=-\sin60^{\circ}=-\frac{\sqrt{3}}{2}$, $\cos300^{\circ}=\cos60^{\circ}=\frac{1}{2}$.

Step3: Calculate $\csc300^{\circ}$

$\csc300^{\circ}=\frac{1}{\sin300^{\circ}}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2\sqrt{3}}{3}$.

Step4: Calculate $\sec300^{\circ}$

$\sec300^{\circ}=\frac{1}{\cos300^{\circ}}=\frac{1}{\frac{1}{2}} = 2$.

Step5: Calculate $\cot300^{\circ}$

$\cot300^{\circ}=\frac{\cos300^{\circ}}{\sin300^{\circ}}=\frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}}=-\frac{\sqrt{3}}{3}$.

Answer:

$\csc300^{\circ}=-\frac{2\sqrt{3}}{3}$, $\sec300^{\circ}=2$, $\cot300^{\circ}=-\frac{\sqrt{3}}{3}$