Question

find the value of each of the six trigonometric functions of the angle θ in the figure.

( sin \theta = )
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals as needed.)

Explanation:

Step1: Find the adjacent side

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). Let the adjacent side be \(a\), opposite side \(b = 9\), hypotenuse \(c = 18\). Then \(a=\sqrt{c^{2}-b^{2}}=\sqrt{18^{2}-9^{2}}=\sqrt{(18 + 9)(18 - 9)}=\sqrt{27\times9}=\sqrt{243}=9\sqrt{3}\).

Step2: Calculate \(\sin\theta\)

The formula for \(\sin\theta\) in a right - triangle is \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, the opposite side to \(\theta\) is \(9\) and the hypotenuse is \(18\). So \(\sin\theta=\frac{9}{18}=\frac{1}{2}\).

Answer:

\(\frac{1}{2}\)