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
35.0
m
(b) the tangent of θ
enter a number. θ is the ratio of two sides of the triangle, but you need to make sure you are expressing the ratio correctly.
(c) the sin of φ
0.385

Explanation:

Part (a)

Step1: Identify triangle type

The triangle is a right - triangle. We know the hypotenuse \(r = 91.0\space m\) and the adjacent side to angle \(\varphi\) is \(y=84.0\space m\). We can use the Pythagorean theorem \(x^{2}+y^{2}=r^{2}\) to find \(x\).

Step2: Apply Pythagorean theorem

Rearranging the formula for \(x\), we get \(x=\sqrt{r^{2}-y^{2}}\). Substitute \(r = 91.0\) and \(y = 84.0\) into the formula:
\[

$$\begin{align*} x&=\sqrt{91.0^{2}-84.0^{2}}\\ &=\sqrt{(91 + 84)(91 - 84)}\\ &=\sqrt{175\times7}\\ &=\sqrt{1225}\\ & = 35.0 \end{align*}$$

\]

Part (b)

Step1: Recall tangent definition

For an angle \(\theta\) in a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For angle \(\theta\), the opposite side is \(x = 35.0\space m\) and the adjacent side is \(y = 84.0\space m\).

Step2: Calculate tangent of \(\theta\)

Using the formula \(\tan\theta=\frac{x}{y}\), substitute \(x = 35.0\) and \(y=84.0\):
\(\tan\theta=\frac{35.0}{84.0}=\frac{35}{84}\approx0.417\)

Part (c)

Step1: Recall sine definition

For an angle \(\varphi\) in a right - triangle, \(\sin\varphi=\frac{\text{opposite}}{\text{hypotenuse}}\). For angle \(\varphi\), the opposite side is \(y = 84.0\space m\) and the hypotenuse is \(r=91.0\space m\).

Step2: Calculate sine of \(\varphi\)

Using the formula \(\sin\varphi=\frac{y}{r}\), substitute \(y = 84.0\) and \(r = 91.0\):
\(\sin\varphi=\frac{84.0}{91.0}=\frac{84}{91}\approx0.923\)

Answer:

35.0