Question

1. a right triangle is shown in the figure. what is the value of \\( \sin \theta \\)?

a. \\( \frac { 5 } { 13 } \\)
b. \\( \frac { 5 } { 12 } \\)
c. \\( \frac { 12 } { 13 } \\)
d. \\( \frac { 13 } { 12 } \\)

Explanation:

Step1: Find the length of the unknown side

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 13\) (hypotenuse) and \(a = 12\). Let the unknown side be \(b\).

$$b=\sqrt{13^{2}-12^{2}}=\sqrt{169 - 144}=\sqrt{25}=5$$

Step2: Calculate \(\sin\theta\)

By the definition of sine in a right - triangle \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). The side opposite to \(\theta\) is \(b = 5\) and the hypotenuse \(c=13\).
\(\sin\theta=\frac{5}{13}\)

Answer:

A. \(\frac{5}{13}\)