Question Video: Finding the Unknown Angles in a Right Triangle Using Trigonometry Mathematics

Video Transcript

For the given figure, find the measures of angle 𝐴𝐶𝐵 and angle 𝐵𝐴𝐶 in degrees to two decimal places.

We’ve been given a right triangle in which we know the lengths of two of its sides. We can therefore approach this problem using trigonometry. Our first step in a problem like this is to label the sides of the triangle. But in order to do this, we need to know which angle we’re labeling the sides in relation to. Let’s calculate angle 𝐴𝐶𝐵 first, and we’ll label this on our diagram as angle 𝑥. The hypotenuse of a right triangle is always the same. It’s the side directly opposite the right angle. The opposite is the side directly opposite the side we’re interested in. So the side opposite angle 𝑥 is the side 𝐴𝐵. And finally, the adjacent is the side between our angle and the right angle. It’s the side 𝐵𝐶.

We can now recall the acronym SOHCAHTOA to help us decide which of the three trigonometric ratios — sine, cosine, or tangent — we need to use to answer this question. The sides whose lengths we’re given are the opposite and adjacent. So we’re going to be using the tan ratio. For an angle 𝜃 in a right triangle, this is defined as the length of the opposite divided by the length of the adjacent. Substituting 𝑥 for the angle 𝜃, four for the opposite, and five for the adjacent, we have the equation tan of 𝑥 is equal to four-fifths.

To determine the value of 𝑥, we need to apply the inverse tangent function, which says if tan of 𝑥 is equal to four-fifths, then what is 𝑥? Evaluating this on our calculators, making sure they’re in degree mode, gives 38.659. We can round this to two decimal places, giving 38.66. So we found the measure of our first angle.

To calculate the final angle in the triangle, we have a choice of methods. We could use trigonometry again, or we could use the fact that the angles in a triangle sum to 180 degrees. It’s more efficient to use the second method. So we have that the measure of angle 𝐵𝐴𝐶 is equal to 180 degrees minus 90 degrees for the right angle minus 38.66 degrees for angle 𝐴𝐶𝐵, which is equal to 51.34 degrees. So we found the measures of the two angles.

If we did want to use trigonometry, we’d have to relabel the sides of the triangle in relation to this angle, which means the opposite and adjacent sides would swap round. We’d still be using the tangent ratio, but this time we’d have tan of 𝑦 is equal to five over four. We’d then have 𝑦 is equal to the inverse tan of five over four, which is indeed equal to 51.34 when rounded to two decimal places.

Our answer to the problem then is that the measure of angle 𝐴𝐶𝐵 is 38.66 degrees and the measure of angle 𝐵𝐴𝐶 is 51.34 degrees, each rounded to two decimal places.