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

๐ด๐ต๐ถ is a right-angled triangle at ๐ต where ๐โ ๐ถ = 62ยฐ and ๐ด๐ถ = 17 cm. Find the lengths of ๐ด๐ต and ๐ต๐ถ giving the answer to two decimal places and the measure of โ ๐ด giving the answer to the nearest degree.

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### Video Transcript

๐ด๐ต๐ถ is a right-angled triangle at ๐ต where the measure of angle ๐ถ is 62 degrees and ๐ด๐ถ is 17 centimeters. Find the lengths of ๐ด๐ต and ๐ต๐ถ giving the answer to two decimal places and the measure of angle ๐ด giving the answer to the nearest degree.

Letโs begin by drawing a sketch of this triangle. Weโre told that it is right-angled at ๐ต. So ๐ต is the vertex by the right angle and the other two vertices are ๐ด and ๐ถ. The other information weโre given is that the measure of angle ๐ถ is 62 degrees and ๐ด๐ถ is 17 centimeters. Weโre asked to find the length of ๐ด๐ต and ๐ต๐ถ. Those are the other two sides of the triangle. So weโll call them ๐ฅ centimeters and ๐ฆ centimeters. And weโll also ask to find the measure of angle ๐ด.

Now, actually, we can work out the measure of angle ๐ด 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, letโs think about how weโre going to find the lengths of the other two sides of this triangle. Weโll begin by labeling all three sides in relation to the angle of 62 degrees. ๐ด๐ถ is the hypotenuse, ๐ด๐ต which weโre calling ๐ฅ centimeters is the opposite, and ๐ต๐ถ is the adjacent.

Weโll then recall the acronym SOHCAHTOA to help us decide which trigonometric ratio we need to calculate the length of each side. Starting with ๐ด๐ต, first of all, the side we want to calculate is the opposite, and the side we know is the hypotenuse. So weโre going to use the sine ratio. This tells us that sin of an angle ๐ is equal to the opposite divided by the hypotenuse. Substituting the values for this triangle, we have sin of 62 degrees is equal to ๐ฅ over 17. We solve for ๐ฅ by multiplying both sides of the equation by 17 giving ๐ฅ equals 17 sin 62 degrees. Evaluating gives 15.0101 which we round to 15.01.

To calculate the second side, ๐ต๐ถ, 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 weโre focusing on trigonometry here, letโs instead calculate ๐ต๐ถ 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 weโre going to use the cosine ratio. Alternatively, we could use the side weโve just calculated, which would give the pair O and A. So weโ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 ๐, ๐ฆ for the adjacent, and 17 for the hypotenuse gives cos of 62 degrees equals ๐ฆ over 17. We can then multiply both sides of the equation by 17 to give ๐ฆ equals 17 cos 62 degrees and evaluate on our calculators, making sure theyโre in degree mode. We then round to two decimal places, giving 7.98. So weโve completed the problem. The length of ๐ด๐ต is 15.01 centimeters. The length of ๐ต๐ถ is 7.98 centimeters, each to two decimal places. And the measure of angle ๐ด is 28 degrees.