Question Video: Using Trigonometry to Solve a Right Triangle Mathematics • 11th Grade

Video Transcript

𝐴𝐵𝐶 is a right triangle at 𝐵, where 𝐵𝐶 equals 10 centimeters and 𝐴𝐶 equals 18 centimeters. Find the length 𝐴𝐵, giving the answer to the nearest centimeter, and the measure of angles 𝐴 and 𝐶, giving the answer to the nearest degree.

Let’s begin by sketching this triangle, which we’re told is a right triangle at 𝐵. The length of 𝐵𝐶 is 10 centimeters and the length of 𝐴𝐶 is 18 centimeters. We need to find the measures of both unknown angles and the length of the third side 𝐴𝐵. Let’s begin by calculating the measure of angle 𝐴, which we can label as 𝜃 on our diagram. As we have a right triangle in which we know two of the side lengths, we can calculate the measure of this angle using right triangle trigonometry. We begin by labeling the three sides of the triangle in relation to this angle. 𝐵𝐶 is the opposite, 𝐴𝐵 is the adjacent, and 𝐴𝐶 is the hypotenuse.

Recalling the acronym SOH CAH TOA, we can see that it is the sine ratio we need to use because the lengths we’ve been given are the opposite and the hypotenuse. Through an angle 𝜃 in a right triangle, the sin of angle 𝜃 is defined to be equal to the length of the opposite divided by the length of the hypotenuse. So for this triangle, we have that sin of 𝜃 is equal to 10 over 18. To find the value of 𝜃, we need to apply the inverse sine function. So we have that 𝜃 is equal to the inverse sin of 10 over 18. Evaluating this on a calculator, which must be in degree mode, gives 33.7489 continuing. And then rounding to the nearest degree gives 34 degrees.

So we found the measure of angle 𝐴. Now let’s consider how we could find the measure of angle 𝐶. If we wish, we could relabel the sides of the triangle in relation to this angle. So 𝐴𝐵 becomes the opposite, and 𝐵𝐶 becomes the adjacent. We could then calculate the measure of angle 𝐶 using the cosine ratio. However, it’s more efficient to recall that angles in any triangle sum to 180 degrees. So to calculate the measure of the third angle in a triangle, we can subtract the measures of the other two angles from 180 degrees. This tells us that angle 𝛼 or angle 𝐶 to the nearest degree is 56 degrees.

Finally, we need to calculate the length of side 𝐴𝐵, which we can do using another trigonometric ratio. In relation to angle 𝜃 or angle 𝐴 whose measure we know, the side 𝐴𝐵 is the adjacent. Using the cosine ratio, we therefore have that the cos of 33.7489 continuing degrees is equal to 𝐴𝐵 over 18. Multiplying both sides of this equation by 18 gives 𝐴𝐵 is equal to 18 cos of 33.7489 degrees. And we’re using the unrounded value here for accuracy. Evaluating this on a calculator gives 14.9666 continuing, and rounding this to the nearest integer gives 15. We have then that the length of 𝐴𝐵 to the nearest centimeter is 15 centimeters. And the measures of angles 𝐴 and 𝐶, each to the nearest degree, are 34 degrees and 56 degrees.

We can check our answer for the length of 𝐴𝐵 using the Pythagorean theorem. In a right triangle, the sum of the squares of the two shorter sides is always equal to the square of the hypotenuse. If we take the unrounded value for 𝐴𝐵 and square it and then add 10 squared for 𝐵𝐶 squared, this gives 324. The square of the hypotenuse, that’s 18 squared, is also equal to 324. And as these two values are the same, this confirms that our answer for 𝐴𝐵 is correct. We could also have calculated the length of 𝐴𝐵 by using the Pythagorean theorem and then checked our answer using trigonometry.