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 with a sketch of this triangle. We’re told that 𝐴𝐵𝐶 is a right-angled triangle at 𝐵. So, 𝐵 is the letter where the right angle is. We’re also told that the measure of angle 𝐶 is 62 degrees and the length of the side 𝐴𝐶, which we see is the hypotenuse of this triangle, is 17 centimeters. We’re asked, first of all, to determine the lengths of 𝐴𝐵 and 𝐵𝐶, which are the two shorter sides in this right-angled triangle. Now, as we haven’t been given the lengths of two sides in this right-angled triangle, we can’t apply the Pythagorean theorem. So, instead, we’re 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 𝐴𝐵𝐶. The side which is directly opposite the right angle — that’s the side 𝐴𝐶 — is called the hypotenuse. The side directly opposite the known angle of 62 degrees — in this case, that’s the side 𝐴𝐵 — is called the opposite. And the side directly between the right angle and the known angle of 62 degrees is called the adjacent. That’s the side 𝐵𝐶. 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 𝐴𝐵, first of all, the two sides involved in this ratio are the opposite and the hypotenuse. So, we’re going to be using SOH. That’s the sine ratio. Let’s recall its definition. Sin of an angle 𝜃 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 that’s the value of 𝜃. The opposite is the side 𝐴𝐵, 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 𝐴𝐵 over 17.
To solve for 𝐴𝐵, we need to multiply both sides of this equation by 17. This leaves just 𝐴𝐵 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 it’s 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 𝐴𝐵 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 we’re rounding down. So, the one in the second decimal place will remain a one. So, we found that the length of the side 𝐴𝐵 is 15.01 centimeters to two decimal places.
Now, let’s consider the length of the side 𝐵𝐶. We have two approaches that we could take. As we’ve worked out the length of 𝐴𝐵, 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 𝐴𝐵. We don’t want to carry this error forward into our calculation of 𝐵𝐶. So, instead, we can use trigonometry again. This time, the two sides involved in our ratio are the adjacent and the hypotenuse. So, we’re going to be using the CAH part of SOHCAHTOA. That’s the cos ratio.
The cosine, or cos, of an angle 𝜃 is defined as the length of the adjacent side divided by the length of the hypotenuse. In our triangle, the angle 𝜃 is 62 degrees and the hypotenuse is 17 centimeters as before. The adjacent is the side 𝐵𝐶. So, we have the equation cos of 62 degrees is equal to 𝐵𝐶 over 17. Just as we did for 𝐴𝐵, we solve this equation by multiplying both sides by 17, giving that 𝐵𝐶 is equal to 17 cos of 62 degrees. Evaluating this on a calculator gives that 𝐵𝐶 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 we’re rounding down. So, the eight in the second decimal place remains an eight. We have, then, that the length of 𝐵𝐶 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 𝐴. Now, in fact, we don’t need to use trigonometry at all in order to find the measure of angle 𝐴 because we have a right-angled triangle in which we know the measures of the two other angles. There’s the right angle of 90 degrees and then the angle at 𝐶 which is 62 degrees.
We can, therefore, work out the measure of angle 𝐴 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 𝐴 is 28 degrees. And we don’t need to round this answer to the nearest degree because it is 28 degrees exactly.
Suppose, though, that we hadn’t noticed this and we had attempted to use trigonometry to answer this question. In relation to this angle at 𝐴, the adjacent and opposite sides of the triangle swap around. The side opposite angle 𝐴 is the side 𝐵𝐶, and the side adjacent to angle 𝐴 is the side 𝐴𝐵. 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 𝐴.
We haven’t used tan yet in this question. So, let’s use the tan ratio to calculate angle 𝐴. The definition of the tan ratio is that tan of an angle 𝜃 is equal to the length of the opposite side divided by the length of the adjacent. The opposite side is the side 𝐵𝐶, which we’ve just calculated to be 7.98 centimeters. And the adjacent side is the side 𝐴𝐵, which we’ve just calculated to be 15.01 centimeters. So, we have that tan of angle 𝐴 is equal to 7.98 over 15.01.
As we’re calculating the measure of an angle, we need to use the inverse trigonometric function inverse tan. So, we have that 𝐴 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 we’re rounding up. So, we round up to 28 degrees, giving the same value for the measure of angle 𝐴 as we found using the angle sum in a triangle.
So, we have, then, that the lengths of 𝐴𝐵 and 𝐵𝐶, each to two decimal places, are 15.01 and 7.98 centimeters. And the measure of angle 𝐴 to the nearest degree is 28 degrees.