### Video Transcript

๐ด๐ต๐ถ is a right-angled triangle at ๐ต, where the measure of angle ๐ถ is 62 degrees and ๐ด๐ถ is 17 centimeters. Find the lengths of ๐ด๐ต and ๐ต๐ถ, giving the answer to two decimal places, and the measure of angle ๐ด, giving the answer to the nearest degree.

Letโs begin with a sketch of this triangle. Weโre told that ๐ด๐ต๐ถ is a right-angled triangle at ๐ต. So, ๐ต is the letter where the right angle is. Weโre also told that the measure of angle ๐ถ is 62 degrees and the length of the side ๐ด๐ถ, which we see is the hypotenuse of this triangle, is 17 centimeters. Weโre asked, first of all, to determine the lengths of ๐ด๐ต and ๐ต๐ถ, which are the two shorter sides in this right-angled triangle. Now, as we havenโt been given the lengths of two sides in this right-angled triangle, we canโt apply the Pythagorean theorem. So, instead, weโre going to apply right-angle trigonometry because we know the measure of one of the other angles in this triangle and the length of one side.

We begin by labeling the three sides of triangle ๐ด๐ต๐ถ. The side which is directly opposite the right angle โ thatโs the side ๐ด๐ถ โ is called the hypotenuse. The side directly opposite the known angle of 62 degrees โ in this case, thatโs the side ๐ด๐ต โ is called the opposite. And the side directly between the right angle and the known angle of 62 degrees is called the adjacent. Thatโs the side ๐ต๐ถ. We can use the memory aid SOHCAHTOA to help us decide which trigonometric ratio we need to use to calculate each of these lengths. Here, S, C, and T stand for sin, cos, and tan. And O, A, and H stand for opposite, adjacent, and hypotenuse.

For ๐ด๐ต, first of all, the two sides involved in this ratio are the opposite and the hypotenuse. So, weโre going to be using SOH. Thatโs the sine ratio. Letโs recall its definition. Sin of an angle ๐ is equal to the length of the opposite side divided by the length of the hypotenuse. In this triangle, the angle we know is 62 degrees, so thatโs the value of ๐. The opposite is the side ๐ด๐ต, which we want to calculate. And the length of the hypotenuse is 17. So, we have the equation sin of 62 degrees is equal to ๐ด๐ต over 17.

To solve for ๐ด๐ต, we need to multiply both sides of this equation by 17. This leaves just ๐ด๐ต on its own on the right-hand side. And on the left-hand side, we have 17 sin of 62 degrees, which we need to calculate it to help us evaluate. We can type this directly into our calculator, making sure that itโs in degree mode. Our calculator returns a value of 15.0101, and then the decimal continues. The question asks us to give our answer for the length of ๐ด๐ต to two decimal places. So, we look at the number in the third decimal place, which is a zero. And as this deciding number is less than five, this tells us that weโre rounding down. So, the one in the second decimal place will remain a one. So, we found that the length of the side ๐ด๐ต is 15.01 centimeters to two decimal places.

Now, letโs consider the length of the side ๐ต๐ถ. We have two approaches that we could take. As weโve worked out the length of ๐ด๐ต, we now know two of the lengths in our right-angled triangle. So we now could apply the Pythagorean theorem to work out the length of the third side. But just suppose we made a mistake in our calculation of ๐ด๐ต. We donโt want to carry this error forward into our calculation of ๐ต๐ถ. So, instead, we can use trigonometry again. This time, the two sides involved in our ratio are the adjacent and the hypotenuse. So, weโre going to be using the CAH part of SOHCAHTOA. Thatโs the cos ratio.

The cosine, or cos, of an angle ๐ is defined as the length of the adjacent side divided by the length of the hypotenuse. In our triangle, the angle ๐ is 62 degrees and the hypotenuse is 17 centimeters as before. The adjacent is the side ๐ต๐ถ. So, we have the equation cos of 62 degrees is equal to ๐ต๐ถ over 17. Just as we did for ๐ด๐ต, we solve this equation by multiplying both sides by 17, giving that ๐ต๐ถ is equal to 17 cos of 62 degrees. Evaluating this on a calculator gives that ๐ต๐ถ is equal to 7.9810 continuing. We are, again, rounding to two decimal places. And the digit in the third decimal place is a one, telling us that weโre rounding down. So, the eight in the second decimal place remains an eight. We have, then, that the length of ๐ต๐ถ to two decimal places is 7.98 centimeters. So, we found the two required lengths.

And now we need to answer the second part of the question, which asks us to calculate the measure of angle ๐ด. Now, in fact, we donโt need to use trigonometry at all in order to find the measure of angle ๐ด because we have a right-angled triangle in which we know the measures of the two other angles. Thereโs the right angle of 90 degrees and then the angle at ๐ถ which is 62 degrees.

We can, therefore, work out the measure of angle ๐ด by subtracting the other two angles in the triangle from 180 degrees, because we know that the angle sum in any triangle is 180 degrees. We have, then, that the measure of angle ๐ด is 28 degrees. And we donโt need to round this answer to the nearest degree because it is 28 degrees exactly.

Suppose, though, that we hadnโt noticed this and we had attempted to use trigonometry to answer this question. In relation to this angle at ๐ด, the adjacent and opposite sides of the triangle swap around. The side opposite angle ๐ด is the side ๐ต๐ถ, and the side adjacent to angle ๐ด is the side ๐ด๐ต. We know the lengths of all three sides in this triangle, so we can use any of the three trigonometric ratios in order to calculate the measure of angle ๐ด.

We havenโt used tan yet in this question. So, letโs use the tan ratio to calculate angle ๐ด. The definition of the tan ratio is that tan of an angle ๐ is equal to the length of the opposite side divided by the length of the adjacent. The opposite side is the side ๐ต๐ถ, which weโve just calculated to be 7.98 centimeters. And the adjacent side is the side ๐ด๐ต, which weโve just calculated to be 15.01 centimeters. So, we have that tan of angle ๐ด is equal to 7.98 over 15.01.

As weโre calculating the measure of an angle, we need to use the inverse trigonometric function inverse tan. So, we have that ๐ด is equal to inverse tan of 7.98 over 15.01. Evaluating this on our calculator, again, making sure our calculator is in degree mode, gives 27.9971 continuing. If we round to the nearest degree, then the nine in the first decimal place tells us that weโre rounding up. So, we round up to 28 degrees, giving the same value for the measure of angle ๐ด as we found using the angle sum in a triangle.

So, we have, then, that the lengths of ๐ด๐ต and ๐ต๐ถ, each to two decimal places, are 15.01 and 7.98 centimeters. And the measure of angle ๐ด to the nearest degree is 28 degrees.