Question

find the value of each of the six trigonometric functions of the angle \\( \theta \\) in the figure
\\( \sin \theta= \\)
(simplify your answer, including any radicals 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 \(b = 1\), and hypotenuse \(c=\sqrt{26}\). Then \(a=\sqrt{c^{2}-b^{2}}=\sqrt{(\sqrt{26})^{2}-1^{2}}=\sqrt{26 - 1}=\sqrt{25}=5\).

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 \(b = 1\) and hypotenuse \(c=\sqrt{26}\). So \(\sin\theta=\frac{1}{\sqrt{26}}\). Rationalize the denominator: \(\sin\theta=\frac{1\times\sqrt{26}}{\sqrt{26}\times\sqrt{26}}=\frac{\sqrt{26}}{26}\).

Answer:

\(\frac{\sqrt{26}}{26}\)