Question

how do I find the measure of the side of a right triangle whose length is designated by the lower case c

Answer 1

The triangle ABC is a right triangle, you know one of the angles, ∠A=31º, the opposite side to the angle, BC=15m, and need to determine the measure of the hypothenuse AB=c

To determine the value of one side of a right triangle, when you know one angle and one side of it you have to use the trigonometric relationships:

`\begin{gathered} \text{sin}\theta=\frac{\text{opposite}}{\text{hypothenuse}} \\ \cos \theta=\frac{\text{adjacent}}{\text{hypothenuse}} \\ \tan \theta=\frac{\text{opposite}}{\text{hypothenuse}} \end{gathered}`

Where

θ indicates one angle of the triangle.

For the exercise, since we know the angle and the opposite side to it, we have to use the sine, because it relates the opposite side with the hypothenuse:

`\begin{gathered} \sin \theta=\frac{opposite}{\text{hypothenuse}} \\ \sin 31º=(15m)/(c) \end{gathered}`

From this expression, we can calculate c as follows

`\begin{gathered} c\cdot\sin 31º=15 \\ c=(15)/(\sin 31º) \\ c=29.12m \end{gathered}`