Question

for the triangle shown in the figure below what are each of the following? (let y = 84.0 m and r = 91.0 m. assume the triangle is a right triangle.) (a) the length of the unknown side x m (b) the tangent of θ (c) the sin of θ

Explanation:

Part (a)

Step1: Identify the triangle type

This is a right - triangle, so we can use the Pythagorean theorem \(r^{2}=x^{2}+y^{2}\), and we want to solve for \(x\). Rearranging the formula gives \(x = \sqrt{r^{2}-y^{2}}\).

Step2: Substitute the values

We know that \(y = 84.0\space m\) and \(r=91.0\space m\). Substitute these values into the formula: \(x=\sqrt{(91.0)^{2}-(84.0)^{2}}\). First, calculate \((91.0)^{2}=8281\) and \((84.0)^{2} = 7056\). Then, \(91.0^{2}-84.0^{2}=8281 - 7056=1225\). Then, \(\sqrt{1225}=35.0\space m\).

Step1: Recall the definition of tangent

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For angle \(\theta\), the opposite side is \(x\) and the adjacent side is \(y\).

Step2: Substitute the values

We found \(x = 35.0\space m\) and \(y = 84.0\space m\). So \(\tan\theta=\frac{x}{y}=\frac{35.0}{84.0}\approx0.4167\).

Step1: Recall the definition of sine

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For angle \(\theta\), the opposite side is \(x\) and the hypotenuse is \(r\).

Step2: Substitute the values

We know \(x = 35.0\space m\) and \(r = 91.0\space m\). So \(\sin\theta=\frac{x}{r}=\frac{35.0}{91.0}\approx0.3846\).

Answer:

\(35.0\space m\)

Part (b)