Question

find the exact values of the six trigonometric functions of the given angle. do not use a calculator.
30°
b. the function value is undefined.
select the correct choice below and fill in any answer boxes within your choice.
a. \\( \csc 30^{\circ}= \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
b. the function value is undefined.
select the correct choice below and fill in any answer boxes within your choice.
a. \\( \sec 30^{\circ}= \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
b. the function value is undefined.
select the correct choice below and fill in any answer boxes within your choice.
a. \\( \cot 30^{\circ}= \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
b. the function value is undefined.

Explanation:

Step1: Recall the definition of $\csc\theta$

$\csc\theta=\frac{1}{\sin\theta}$. For $\theta = 30^{\circ}$, $\sin30^{\circ}=\frac{1}{2}$. So, $\csc30^{\circ}=\frac{1}{\sin30^{\circ}}=\frac{1}{\frac{1}{2}} = 2$.

Step2: Recall the definition of $\sec\theta$

$\sec\theta=\frac{1}{\cos\theta}$. For $\theta = 30^{\circ}$, $\cos30^{\circ}=\frac{\sqrt{3}}{2}$. So, $\sec30^{\circ}=\frac{1}{\cos30^{\circ}}=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}$.

Step3: Recall the definition of $\cot\theta$

$\cot\theta=\frac{\cos\theta}{\sin\theta}$. For $\theta = 30^{\circ}$, $\sin30^{\circ}=\frac{1}{2}$ and $\cos30^{\circ}=\frac{\sqrt{3}}{2}$. So, $\cot30^{\circ}=\frac{\cos30^{\circ}}{\sin30^{\circ}}=\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\sqrt{3}$.

Answer:

• For $\csc30^{\circ}$: A. $\csc30^{\circ}=2$
• For $\sec30^{\circ}$: A. $\sec30^{\circ}=\frac{2\sqrt{3}}{3}$
• For $\cot30^{\circ}$: A. $\cot30^{\circ}=\sqrt{3}$