Question

find the exact values of the six trigonometric functions of the given angle. do not use a calculator.
30°
select the correct choice below and fill in any answer boxes within your choice.
oa. sin 30°=
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
ob. the function value is undefined.
select the correct choice below and fill in any answer boxes within your choice.
oa. cos 30°=
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
ob. the function value is undefined.
select the correct choice below and fill in any answer boxes within your choice.
oa. tan 30°=
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
ob. the function value is undefined.
select the correct choice below and fill in any answer boxes within your choice.

Explanation:

Step1: Recall the values of trigonometric functions for \(30^{\circ}\)

We know from the unit - circle or special right - triangle (a \(30 - 60-90\) triangle with sides in the ratio \(1:\sqrt{3}:2\)):

• For \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), in a \(30 - 60 - 90\) triangle, if the hypotenuse \(c = 2\) and the side opposite to the \(30^{\circ}\) angle \(a=1\), then \(\sin30^{\circ}=\frac{1}{2}\)
• For \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), the side adjacent to the \(30^{\circ}\) angle \(b = \sqrt{3}\), then \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\)
• For \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), \(\tan30^{\circ}=\frac{\sin30^{\circ}}{\cos30^{\circ}}=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\)

Answer:

• For \(\sin30^{\circ}\): A. \(\sin30^{\circ}=\frac{1}{2}\)
• For \(\cos30^{\circ}\): A. \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\)
• For \(\tan30^{\circ}\): A. \(\tan30^{\circ}=\frac{\sqrt{3}}{3}\)