Question Video: Using the Sine Rule to Find the Angles of a Triangle Physics • 9th Grade

What is the size of angle 𝐴, in degrees, in the triangle shown?


Video Transcript

What is the size of angle 𝐴, in degrees, in the triangle shown?

In this triangle, we see the angle 𝐴 marked out here, as well as the interior angle of 54 degrees. Opposite this 54-degree angle is a side length of 8.4 centimeters. And then opposite angle 𝐴 is a side length of 9.6 centimeters. We see then that this triangle is well set up for us to use the sine rule to solve for this angle 𝐴. The sine rule applies to any triangle. It says that if the interior angles and the corresponding side lengths are marked out as shown, then the sine of any of those interior angles divided by the corresponding side length is equal to that same ratio for any of the other pairs of angles and sides.

So as we think about applying the sine rule to our triangle over here, specifically to solve for angle 𝐴, we remind ourselves that angle 𝐴 corresponds to the side length of 9.6 centimeters and the angle of 54 degrees corresponds to 8.4 centimeters. So then the ratio of the sin of the unknown angle 𝐴 to 9.6 centimeters is equal, by the sine rule, to the sin of 54 degrees divided by 8.4 centimeters.

And now what we want to do is to isolate this angle 𝐴. To do this, we’ll first multiply both sides of the equation by 9.6 centimeters, canceling that factor on the left and then giving us this expression here. Notice that on the right-hand side of our equation, these units of centimeters cancel from numerator and denominator.

And now what we need to do is to invert or undo the application of the sine function on the angle 𝐴 we want to solve for. We do this by applying what’s called the arc sine or the inverse sine to the sin of 𝐴. And then to maintain our equality, we apply the same inverse sine function to the right-hand side of our expression. When we take the inverse sine of the sin of the angle 𝐴, what remains is simply the angle 𝐴.

Our final step is to evaluate the right-hand side of this expression. And just as the sine function is standard on any scientific calculator, so the inverse sine function will be as well. When we evaluate this expression and find our answer in degrees, to two significant figures, it’s 68 degrees. That’s the size of the angle 𝐴 in the triangle shown.

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