Question

if, in a right triangle, $\theta = 14^{circ}$ and the side opposite to $\theta$ has length 5, how long is the hypotenuse? round to at least 3 decimal places.

Explanation:

Step1: Recall the sine formula

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Let the hypotenuse be \(h\). Given \(\theta = 14^{\circ}\) and the opposite side \(= 5\). So, \(\sin(14^{\circ})=\frac{5}{h}\).

Step2: Solve for \(h\)

We can rewrite the formula as \(h=\frac{5}{\sin(14^{\circ})}\). Since \(\sin(14^{\circ})\approx0.2419\) (using a calculator), then \(h = \frac{5}{0.2419}\).

Step3: Calculate the value of \(h\)

\(h=\frac{5}{0.2419}\approx20.669\)

Answer:

\(20.669\)