Question

a right triangle with one leg length 9, hypotenuse length m, and one acute angle 5 degrees. find the value of m.

Explanation:

Step1: Identify trigonometric ratio

We have a right triangle, and we know the opposite side to the \(5^\circ\) angle is 9, and \(m\) is the hypotenuse. The sine function relates the opposite side and the hypotenuse: \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\).
So, \(\sin(5^\circ)=\frac{9}{m}\).

Step2: Solve for \(m\)

Rearrange the formula to solve for \(m\): \(m = \frac{9}{\sin(5^\circ)}\).
Calculate \(\sin(5^\circ)\approx0.08716\).
Then \(m=\frac{9}{0.08716}\approx103.26\).

Answer:

\(m\approx103.26\) (or more precise value depending on calculator precision)