Video: Using the Law of Sines to Calculate an Unknown Length

For the given figure, 𝐵𝐶 = 7, 𝑚∠𝐴𝐵𝐶 = 110°, and 𝑚∠𝐵𝐴𝐶 = 38°. Work out the length of 𝐴𝐶. Give your answer to two decimal places.

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Video Transcript

For the given figure, 𝐵𝐶 equals seven, the angle 𝐴𝐵𝐶 is equal to 110 degrees, and the angle 𝐵𝐴𝐶 is equal to 38 degrees. Work out the length of 𝐴𝐶. Give your answer to two decimal places.

The first thing we’re going to do to solve this problem is actually to annotate the diagram using the information from the question. Well, first of all, we have that 𝐵𝐶 equals seven. So I’ve marked that on. Next, we actually have that angle 𝐴𝐵𝐶 is equal to 110 degrees. So I’ve marked that on. And then finally, we have angle 𝐵𝐴𝐶 which equals 38 degrees.

Okay, great. So we’ve marked on the information that we’ve got. I’ve also marked on 𝐴𝐶 because that’s what we’re looking to find. Okay, great. So we’ve got all the information. Now let’s get on with solving the problem. So before we can actually get on and calculate the value of 𝐴𝐶, what we need to do is decide how we’re going to solve it. And to do that, I ask a couple of questions first of all.

First question is: does it have a right angle? We can see this triangle doesn’t have a right angle. So therefore, we know it’s not going to be the trigonometric ratios or it’s gonna be the Pythagorean theorem. What we’re actually gonna be using is either the sine or the cosine rule. Then the next question I ask is: does it have two matching pairs? By two matching pairs, what I mean is the angle and the side opposite them. Well, actually, if it’s yes, then it’ll be the sine rule. If it’s no, then it’s the cosine rule. Well, in this question, we have angle 𝐵𝐴𝐶 and the side opposite it. So that’s one matching pair. Then we have angle 𝐴𝐵𝐶. And we want to find the side opposite it. So yes, that would be another matching pair that we’re going to be using. So the answer is: yes. So therefore, we know we’re gonna use the sine rule.

Okay, great. Let’s get on and solve the problem. Well, the sine rule actually tells us that 𝑎 over sign 𝐴 is equal to 𝑏 over sign 𝐵 is equal to 𝑐 over sign 𝐶. So that’s what we’re gonna be using to solve this problem. You can actually also have the sine rule the other way round where it be sine 𝑎 over 𝐴, sine 𝑏 over 𝐵, and sine 𝑐 over 𝐶. But we’re gonna use it this way round because we’re looking to find the length of a side and not an angle.

The first thing we’re gonna do is actually substitute values in because what we have is a side divided by the sign of the corresponding angle equals another side divided by the sine of its corresponding angle. So what we have is 𝐴𝐶 over that sine 110 degrees, and that’s because that’s the angle opposite 𝐴𝐶, is equal to seven over sine of 38 degrees, again because that’s the angle opposite to seven.

And now, what we’re gonna do is we’re actually gonna solve this equation to find 𝐴𝐶. And to do that, first thing we’re gonna do is multiply both sides by sine of 110 degrees which is gonna give us 𝐴𝐶 is equal to seven sine 110 degrees divided by sine 38 degrees. And then, when we calculate this, we get 𝐴𝐶 is equal to 10.684196769. So then, one more stage to do to complete the question because the question actually asks us to round to two decimal places. So that’s what we’re gonna do.

So therefore, we can say that the length of 𝐴C is equal to 10.68, to two decimal places.

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