There are two buttons on your graphing calculator that help you solve problems involving triangles: the inverse sine and inverse cosine buttons. Click the image below to see where these buttons are located on most calculators.
We can try out the inverse sine button using our swimming example. Earlier, you figured the distance you swam from one end of the lake to the other was 340.66 meters. You knew that \(\small\mathsf{ {AB} }\) was 125 meters, and let's say you know that \(\small\mathsf{ \angle }\)C is 14.1° though you don't know the measurement of \(\small\mathsf{ \angle }\)A. Can we use the Law of Sines to determine the measurement of \(\small\mathsf{ \angle }\)A?
Identify the information you are given:
\(\small\mathsf{ {AB} }\) = 125
\(\small\mathsf{ {BC} }\) = 340.66
\(\small\mathsf{ \angle }\)C = 14.1°
Now use the Law of Sines to write a proportion:
\(\mathsf{ \frac{125}{sin14.1°} }\) = \(\mathsf{ \frac{340.66}{sin{\angle}A} }\)
or
125(sin\(\small\mathsf{ \angle }\)A) = 340.66(sin14.1°)
then
\(\mathsf{sin{ \angle }A = \frac{340.66(sin14.1°)}{125} }\)
sin\(\small\mathsf{ \angle }\)A = .6639
