### Video Transcript

For the given figure, π΄π΅ is equal to three and π΅πΆ is equal to π. Use the law of sines to work out π. Give your answer to two decimal places.

Letβs begin by adding the given measurements to our triangle. We can see that we have a nonright-angled triangle for which we know the measure of two angles and the length of one side. To find the length of the side labelled π, weβll need to use the law of sines: π over sin π΄ equals π over sin π΅, which equals π over sin πΆ.

Alternatively, that can be written as sin π΄ over π equals sin π΅ over π, which equals sin πΆ over π. We only need to use one of these forms. since weβre trying to calculate the length of one of the sides, weβll use the first form. It doesnβt particularly matter either way. But by using the first form here, it will minimize the amount of rearranging weβll need to do to solve the equation.

Next, weβll label the sides of the triangle. The side opposite the angle π΄ is already given by the lowercase π. The side opposite the angle π΅ is lowercase π, and the side opposite the angle πΆ is lowercase π. For the law of sines, we usually only need to use two parts. We donβt know the side labelled lowercase π, so weβre going to use π over sin π΄ and π over sin πΆ.

Letβs substitute what we know into this formula. That gives us π over sin 64 is equal to three over sin 31. To solve this equation and work out the value of π, weβll need to multiply both sides by sine of 64. π is therefore equal to three over sine of 31 multiplied by sine of 64.

If we put that into our calculator, we get that π is equal to 5.2353 and so on. Correct to two decimal places, π is equal to 5.24. Notice how there were no units provided in the question, so no units are required in our answer.