Question

question
in $\triangle ghi$, $i = 3.6$ cm, $h = 3.4$ cm and $\angle h = 112^\circ$. find all possible values of $\angle i$, to the nearest 10th of a degree.
answer attempt 2 out of 2

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that $\frac{\sin I}{i}=\frac{\sin H}{h}$. We know $i = 3.6$, $h = 3.4$, and $\angle H=112^\circ$. Plugging in the values:
$\frac{\sin I}{3.6}=\frac{\sin 112^\circ}{3.4}$

Step2: Solve for $\sin I$

First, calculate $\sin 112^\circ\approx\sin(90^\circ + 22^\circ)=\cos 22^\circ\approx0.9272$. Then:
$\sin I=\frac{3.6\times\sin 112^\circ}{3.4}=\frac{3.6\times0.9272}{3.4}\approx\frac{3.33792}{3.4}\approx0.9817$

Step3: Find the reference angle

We find the reference angle by taking the inverse sine: $\arcsin(0.9817)\approx79.0^\circ$. Now, we need to check if there is another possible angle. Since $\angle H = 112^\circ$ is obtuse, the other angle $I$ must be acute (because the sum of angles in a triangle is $180^\circ$, and we can't have two obtuse angles). So we only consider the acute angle.

Answer:

$79.0^\circ$