Find Side Length using Law of Sines in a Triangle

Published on March 24, 2026

Trigonometry

Calculate the unknown side 'e' of a triangle given two angles (47 degrees and 54 degrees) and one non-included side (length 12) by first finding the third angle and then applying the Law of Sines.

Question

Answer

Here is the step-by-step solution to find the value of

.

The image shows a triangle

with the following known values:

Angle at , .
Angle at , .
Side opposite to , .
Side opposite to , .
Side opposite to , . Wait, looking closely at the image, the side with length 12 is , which is opposite to . Let’s denote the side lengths by their opposite angles.
- Side is (opposite ).
- Side is (opposite ). We are given .
- Side is (opposite ).

We are given two angles and the side included between them (

) is NOT opposite to the unknown angle

, nor is it opposite to one of the known angles. We are given two angles and one side that is NOT the third side (the side opposite the third angle).

Given:

Side (This side is opposite to ).
We need to find side (This side is opposite to ).

Step 1: Find the third angle, .
The sum of the angles in a triangle is

.

Step 2: Use the Law of Sines to find .
The Law of Sines states that for a triangle with sides

and opposite angles

:

In

:

We want to find

, and we know

and its opposite angle

. We also know

.

Step 3: Solve for .

Now, we calculate the values of the sine functions:

Step 4: Round the answer to the nearest tenth.
Rounding

to the nearest tenth gives

.

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