Video: Using the Sine Rule to Find the Size of an Angle of a Triangle

๐ด๐ต๐ถ is a triangle where ๐‘šโˆ ๐ด = 138ยฐ, ๐‘Ž = 13 cm and ๐‘ = 7 cm. Find ๐‘šโˆ ๐ต giving the answer to the nearest second.

04:12

Video Transcript

๐ด๐ต๐ถ is a triangle, where the measure of the angle ๐ด is 138 degrees, ๐‘Ž is 13 centimeters, and ๐‘ is seven centimeters. Find the measure of the angle at ๐ต, giving the answer to the nearest second.

Letโ€™s begin by sketching a diagram of this triangle. Remember the diagram doesnโ€™t need to be to scale, but it should be roughly in proportion so we can check the validity of any answers we get. Here we have a nonright-angled triangle for which we know the length of two sides and one of the angles. Weโ€™ll need to use the law of sines to find the measure of the angle at ๐ต. Remember to use the law of cosines, we would need a minimum of two sides plus an included angle.

In this triangle, the included angle would be the angle at ๐ถ, which we donโ€™t know. The law of sines says that sin ๐ด over ๐‘Ž equals sin ๐ต over ๐‘ which equals sin ๐ถ over ๐‘. Alternatively, we could write that as ๐‘Ž over sin ๐ด, which equals ๐‘ over sin ๐ต, which equals ๐‘ over sin ๐ถ. We can use either of these formulae. In fact, we choose to use the first formula because it will minimize the amount of rearranging weโ€™ll need to do to solve our equation.

The second formula is more useful when youโ€™re trying to calculate the length of a missing side. Remember, for the law of sines, we need pairs of angles and their associated side. Here we know the measure of the angle at ๐ด and the length of the side ๐‘Ž. We also know the measure of the angle at ๐ต, and we know the length of the side ๐‘. Since we donโ€™t know anything about the side or the angle ๐ถ, weโ€™ll use sin ๐ด over ๐‘Ž equals sin ๐ต over ๐‘.

Substituting the values from our triangle into the formula gives us sine of 138 degrees over 13 is equal to sine of ๐‘ฅ over seven. To solve, weโ€™ll multiply both sides by seven. That gives us sine of ๐‘ฅ is equal to seven sine of 138 degrees all over 13. Thatโ€™s equal to 0.360. Next, weโ€™ll find the inverse sine of both sides of this equation. That gives us that ๐‘ฅ is equal to the inverse sine of 0.360, which is 21.1186 degrees.

Remember though, the question is asking us to leave our answer correct to the nearest second. We will need to convert from degrees into minutes and seconds. Thatโ€™s base 60. We know the integer part of this number tells us that there are 21 degrees. So letโ€™s deal with the decimal part. If we subtract 21 from our original number, weโ€™re left with purely the decimal part. Thatโ€™s 0.118 and so on. We can multiply that by 60 to give us 7.121.

The integer part of this number tells us the number of minutes. Currently then, we have 21 degrees and seven minutes. Letโ€™s then deal with the decimal part of 7.121. If we subtract seven from this decimal, weโ€™re left with 0.121 and so on. And we can multiply that by 60 to tell us the number of seconds. The integer part of this number is the number of seconds. And since itโ€™s correct to the nearest second and the deciding number is two, which is less than five, this rounds down to seven.

The measure of the angle at ๐ต correct to the nearest second is 21 degrees, seven minutes, and seven seconds. Some calculators do have a button that will convert an answer into minutes and seconds for us. It looks like this! And simply pressing this button after calculating the answer of 21.118 and so on will convert it to 21 degrees, seven minutes, and seven seconds. Thatโ€™ll save us a little bit of time.

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