# Question Video: Using Trigonometry to Solve Right-Angled Triangles with Angles in Degrees Mathematics

𝐴𝐵𝐶 is a right-angled triangle at 𝐵 where 𝑚∠𝐶 = 62° and 𝐴𝐶 = 17 cm. Find the lengths of 𝐴𝐵 and 𝐵𝐶 giving the answer to two decimal places and the measure of ∠𝐴 giving the answer to the nearest degree.

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### Video Transcript

𝐴𝐵𝐶 is a right-angled triangle at 𝐵 where the measure of angle 𝐶 is 62 degrees and 𝐴𝐶 is 17 centimeters. Find the lengths of 𝐴𝐵 and 𝐵𝐶 giving the answer to two decimal places and the measure of angle 𝐴 giving the answer to the nearest degree.

Let’s begin by drawing a sketch of this triangle. We’re told that it is right-angled at 𝐵. So 𝐵 is the vertex by the right angle and the other two vertices are 𝐴 and 𝐶. The other information we’re given is that the measure of angle 𝐶 is 62 degrees and 𝐴𝐶 is 17 centimeters. We’re asked to find the length of 𝐴𝐵 and 𝐵𝐶. Those are the other two sides of the triangle. So we’ll call them 𝑥 centimeters and 𝑦 centimeters. And we’ll also ask to find the measure of angle 𝐴.

Now, actually, we can work out the measure of angle 𝐴 straightaway because we have a triangle in which we know the other two angles. The angle sum in any triangle is 180 degrees, so we can work out the measure of the third angle by subtracting the other two from 180 degrees. That gives 28 degrees. Now, let’s think about how we’re going to find the lengths of the other two sides of this triangle. We’ll begin by labeling all three sides in relation to the angle of 62 degrees. 𝐴𝐶 is the hypotenuse, 𝐴𝐵 which we’re calling 𝑥 centimeters is the opposite, and 𝐵𝐶 is the adjacent.

We’ll then recall the acronym SOHCAHTOA to help us decide which trigonometric ratio we need to calculate the length of each side. Starting with 𝐴𝐵, first of all, the side we want to calculate is the opposite, and the side we know is the hypotenuse. So we’re going to use the sine ratio. This tells us that sin of an angle 𝜃 is equal to the opposite divided by the hypotenuse. Substituting the values for this triangle, we have sin of 62 degrees is equal to 𝑥 over 17. We solve for 𝑥 by multiplying both sides of the equation by 17 giving 𝑥 equals 17 sin 62 degrees. Evaluating gives 15.0101 which we round to 15.01.

To calculate the second side, 𝐵𝐶, we have a choice. As we now know the length of two sides in this right triangle, we could calculate the length of the third side by applying the Pythagorean theorem. But as we’re focusing on trigonometry here, let’s instead calculate 𝐵𝐶 using the trigonometric ratios. This time, the side we want to calculate is the adjacent and the side we were originally given is the hypotenuse. So we’re going to use the cosine ratio. Alternatively, we could use the side we’ve just calculated, which would give the pair O and A. So we’d be using the tan ratio. But it makes sense to use the value we were originally given in case you made any mistakes when calculating the length of the opposite.

Substituting 62 degrees for 𝜃, 𝑦 for the adjacent, and 17 for the hypotenuse gives cos of 62 degrees equals 𝑦 over 17. We can then multiply both sides of the equation by 17 to give 𝑦 equals 17 cos 62 degrees and evaluate on our calculators, making sure they’re in degree mode. We then round to two decimal places, giving 7.98. So we’ve completed the problem. The length of 𝐴𝐵 is 15.01 centimeters. The length of 𝐵𝐶 is 7.98 centimeters, each to two decimal places. And the measure of angle 𝐴 is 28 degrees.