Question

use the table to approximate gh in the triangle below. choose 1 answer: a 2.14 units b 3.3 units c 6.4 units d 7.1 units

Explanation:

Step1: Identify the trig - ratio

In right - triangle $GHI$ with right - angle at $H$, we know the adjacent side to angle $I$ is $GH$ and the hypotenuse is $GI = 3$. We use the cosine function since $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 65^{\circ}$, and $\cos(65^{\circ})=\frac{GH}{GI}$.

Step2: Solve for $GH$

We know that $GI = 3$ and $\cos(65^{\circ})\approx0.423$. Substituting the values into the formula $\cos(65^{\circ})=\frac{GH}{3}$, we get $GH = 3\times\cos(65^{\circ})$.
$GH=3\times0.423 = 1.269\approx1.27$ (This is wrong. Let's use tangent instead).
In right - triangle $GHI$ with right - angle at $H$, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta = 65^{\circ}$, the opposite side to angle $I$ is $GH$ and the adjacent side is $HI = 3$. So $\tan(65^{\circ})=\frac{GH}{3}$.
Since $\tan(65^{\circ})\approx2.145$, then $GH = 3\times\tan(65^{\circ})$.
$GH=3\times2.145 = 6.435\approx6.4$

Answer:

C. 6.4 units