Video: Using the Sine Rule to Find the Side Lengths of a Triangle

What is the length of side π‘Ž of the triangle in the diagram?

02:53

Video Transcript

What is the length of side π‘Ž of the triangle shown in the diagram?

In this diagram, we see side π‘Ž right here, as well as two interior angles of this triangle, 64 degrees and 38 degrees. And then opposite this angle of 38 degrees, we have a side length given as 6.1 centimeters. Knowing this, we want to solve for the side length π‘Ž. And we’re going to do it by applying what’s called the sine rule. This rule says that if we have a triangle, and it could be any triangle, with its angles and sides marked out like this, then the ratio of the sine of any of these angles to the corresponding side length is equal to that same ratio for the other pairs of sides and angles.

We can see that a key to being able to use the sine rule is to have at least two corresponding angle and side pairs. That way, we can set up an equality and then solve for any one of those four values, given the other three. As we look to apply the sine rule to our diagram in order to solve for the side length π‘Ž, we see that we have one such angle and side pair, 38 degrees with 6.1 centimeters, but that that’s the only one. We don’t have a second corresponding pair. If we knew what this angle of the triangle was though, we see that that angle is opposite the side length π‘Ž we want to solve for.

It turns out that we can solve for this angle, but we won’t do it using the sine rule. Instead, we’ll use the fact that the sum of the interior angles of any triangle is always 180 degrees. So therefore, if we call this unknown angle in our triangle capital 𝐴, then we can say that 𝐴 plus 64 degrees plus 38 degrees is equal to 180 degrees. Now, 64 degrees plus 38 degrees is 102 degrees. And if we subtract this amount from the left-hand side, the negative 102 cancels with positive 102. And we find that 𝐴 equals 180 degrees minus 102 degrees, or 78 degrees.

Now that we know this angle in our triangle, we can use the sine rule to solve for the side length π‘Ž. We’ll write that the sin of capital 𝐴, which we know is 78 degrees, divided by the side length lowercase π‘Ž is equal to this ratio over here, the sine of this angle in our triangle divided by this side length. Solving for π‘Ž is now just a matter of cross multiplying. We can start by multiplying both sides of the equation by π‘Ž, leading that factor to cancel on the left. And then, in our resulting equation, if we want to isolate π‘Ž on the right-hand side, we do this by multiplying both sides by the inverse of this ratio. When we multiply this ratio by its inverse, that product is equal to one, effectively canceling all this out. And finally, we have an expression we can enter in on our calculator to solve for π‘Ž. To two significant figures, π‘Ž is 9.7 centimeters. That’s the length of this side of the triangle.

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