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Question Video: Using the Law of Sines to Calculate an Unknown Length Mathematics • 11th Grade

For the given figure, ๐ด๐ต = 3 and ๐ต๐ถ = ๐‘Ž. Use the law of sines to work out ๐‘Ž. Give your answer to two decimal places.


Video Transcript

For the given figure, ๐ด๐ต is equal to three and ๐ต๐ถ is equal to ๐‘Ž. Use the law of sines to work out ๐‘Ž. Give your answer to two decimal places.

Letโ€™s 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 ๐‘Ž, weโ€™ll need to use the law of sines: ๐‘Ž over sin ๐ด equals ๐‘ over sin ๐ต, which equals ๐‘ over sin ๐ถ.

Alternatively, that can be written as sin ๐ด over ๐‘Ž equals sin ๐ต over ๐‘, which equals sin ๐ถ over ๐‘. We only need to use one of these forms. Since weโ€™re trying to calculate the length of one of the sides, weโ€™ll use the first form. It doesnโ€™t particularly matter either way. But by using the first form here, it will minimize the amount of rearranging weโ€™ll need to do to solve the equation.

Next, weโ€™ll label the sides of the triangle. The side opposite the angle ๐ด is already given by the lowercase ๐‘Ž. The side opposite the angle ๐ต is lowercase ๐‘, and the side opposite the angle ๐ถ is lowercase ๐‘. For the law of sines, we usually only need to use two parts. We donโ€™t know the side labelled lowercase ๐‘, so weโ€™re going to use ๐‘Ž over sin ๐ด and ๐‘ over sin ๐ถ.

Letโ€™s substitute what we know into this formula. That gives us ๐‘Ž over sin 64 is equal to three over sin 31. To solve this equation and work out the value of ๐‘Ž, weโ€™ll need to multiply both sides by sine of 64. ๐‘Ž is therefore equal to three over sine of 31 multiplied by sine of 64.

If we put that into our calculator, we get that ๐‘Ž is equal to 5.2353 and so on. Correct to two decimal places, ๐‘Ž is equal to 5.24. Notice how there were no units provided in the question, so no units are required in our answer.

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