Question

8.7.2 trigonometric ratios in special triangles. suppose that the earth, the sun, and jupiter are aligned so that the following right triangle is formed. use the information to find the distance between jupiter and the sun, and to find the distance at this particular time from earth to jupiter. round all results to two decimal places, as needed. 1) what is the distance between jupiter and the sun, in millions of miles? round all results to two decimal places million miles 2) what is the distance between earth and jupiter, in millions of miles? round all results to two decimal places million miles

Explanation:

Step1: Find the distance between Jupiter and the Sun ($j$)

We know that in a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta = 83.2^{\circ}$, the opposite side is $j$ (distance between Jupiter and the Sun), and the adjacent side is $93$ (distance between Earth and the Sun).
So, $\tan(83.2^{\circ})=\frac{j}{93}$.
Since $\tan(83.2^{\circ})\approx8.26$, then $j = 93\times\tan(83.2^{\circ})$.
$j=93\times8.26 = 768.18$

Step2: Find the distance between Earth and Jupiter ($e$)

We know that in a right - triangle, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 83.2^{\circ}$, the adjacent side is $93$, and the hypotenuse is $e$ (distance between Earth and Jupiter).
So, $\cos(83.2^{\circ})=\frac{93}{e}$.
Since $\cos(83.2^{\circ})\approx0.12$, then $e=\frac{93}{\cos(83.2^{\circ})}$.
$e=\frac{93}{0.12}=775.00$

Answer:

1. $768.18$ million miles
2. $775.00$ million miles