### Video Transcript

In the figure, ๐ด๐ถ is equal to 3.5. What is ๐ด๐ต? Give your answer to two decimal places.

Letโs begin by adding the length ๐ด๐ถ to our diagram. Now weโre trying to find the length of the side ๐ด๐ต, 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 ๐ด๐ท, before using it again to find the side ๐ด๐ต. 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. Thatโs to say, we know the length of the side ๐ and the measure of the angle at ๐ต and we know the measure of the angle at ๐ถ, and weโre trying to find the length of the side ๐.

We can use either form for the law of sines. However, since weโre trying to find a missing length, itโs sensible to use the first version. This will minimise the amount of rearranging we need to do. Similarly, if weโre trying to find a missing angle, weโd use the second version of the formula.

We know neither the measure of the angle at ๐ด nor the length ๐. So weโre going to use these two parts of the formula: ๐ over sin ๐ต equals ๐ over sin ๐ถ. Substituting each of these measurements into our formula gives us 3.5 over sin of 63 equals ๐ 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 ๐ด๐ต is 2.58 units.