Question

a right triangle has one side with length ( b = 22.27 ) cm and an angle ( \theta = 26.3^{circ} ). find the length of the hypotenuse ( c ) in centimeters. report your answer to two decimal places. note: triangle is not drawn to scale. also note that, because the angle is given in degrees, your calculator should be in degrees.

Explanation:

Step1: Recall the sine formula

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, the side \(b\) is opposite to the angle \(\theta\), and the hypotenuse is \(c\). So, \(\sin\theta=\frac{b}{c}\).

Step2: Rearrange the formula to solve for \(c\)

From \(\sin\theta=\frac{b}{c}\), we can get \(c = \frac{b}{\sin\theta}\).

Step3: Substitute the given values

Given \(b = 22.27\) cm and \(\theta=26.3^{\circ}\). Then \(c=\frac{22.27}{\sin(26.3^{\circ})}\).
We know that \(\sin(26.3^{\circ})\approx0.443\).
So, \(c=\frac{22.27}{0.443}\).

Step4: Calculate the value of \(c\)

\(c=\frac{22.27}{0.443}\approx50.27\)

Answer:

\(50.27\)