In the figure, 𝐴𝐶 is equal to 3.5. What is 𝐴𝐵? Give your answer to two decimal places.
Let’s begin by adding the length 𝐴𝐶 to our diagram. Now we’re trying to find the length of the side 𝐴𝐵, and there are two ways that we can do this. Notice that the triangle split into two right-angled triangles.
The first method we could use right angle trigonometry to calculate the length of the side 𝐴𝐷, before using it again to find the side 𝐴𝐵. However, if we look carefully, we can see that we have a non-right-angled triangle for which we know the length of one side and the measure of two of its angles. This means we can use the law of sines to calculate the missing length.
The law of sines requires the triangle to have pairs of sides and angles. That’s to say, we know the length of the side 𝑏 and the measure of the angle at 𝐵 and we know the measure of the angle at 𝐶, and we’re trying to find the length of the side 𝑐.
We can use either form for the law of sines. However, since we’re trying to find a missing length, it’s sensible to use the first version. This will minimise the amount of rearranging we need to do. Similarly, if we’re trying to find a missing angle, we’d use the second version of the formula.
We know neither the measure of the angle at 𝐴 nor the length 𝑎. So we’re going to use these two parts of the formula: 𝑏 over sin 𝐵 equals 𝑐 over sin 𝐶. Substituting each of these measurements into our formula gives us 3.5 over sin of 63 equals 𝑐 over sin of 41.
To solve the equation, we can multiply both sides by sin of 41. That gives us 3.5 over sin of 63 multiplied by sin of 41. Typing that into our calculator gives us 2.5770. Correct to two decimal places, the length of 𝐴𝐵 is 2.58 units.