Lesson Video: Right Triangle Trigonometry: Solving for an Angle Mathematics • 11th Grade

Video Transcript

In this video, we will learn how to find the missing angle in a right triangle using the appropriate inverse trigonometric function, given two side lengths.

Let’s begin by recapping some of the vocabulary related to right triangles. Suppose we have a right triangle with one of the non-right angles labeled as 𝜃. The sides of the triangle have specific names in relation to this angle. The side of the triangle, which is directly opposite the right angle, which is always the longest side of a triangle, is called the hypotenuse. In relation to angle 𝜃, the side directly opposite this angle is called the opposite. And the final side of the triangle, which is between angle 𝜃 and the right angle, is called the adjacent. We’ll often see the names for these three sides abbreviated to opp., adj., and hyp. or simply O, A, and H.

The three trigonometric ratios sine, cosine, and tangent describe the ratios between different pairs of side lengths in a right triangle. For a fixed value of 𝜃, the ratio between each pair of side lengths is always the same, no matter how big the triangle is. We can use the acronym SOH CAH TOA to remember the definitions of the three trigonometric ratios. The first letter in each part refers to the trigonometric ratio either sine, cosine, or tangent. And then the next two letters refer to the pair of sides involved in that particular ratio, with the side in the numerator first, followed by the side in the denominator. So sin 𝜃 is equal to the opposite divided by the hypotenuse. cos of 𝜃 is the length of the adjacent divided by the length of the hypotenuse. And tangent or tan of 𝜃 is the opposite divided by the adjacent.

We should already be comfortable with using these three trigonometric ratios to calculate the length of a side in a right triangle, given the length of one of the other two sides and the measure of one of the non-right angles. In this video, we’ll focus on finding the measure of an angle given the length of two of the triangle’s sides. To do this, we’ll need to use the inverse trigonometric functions. These are essentially the functions that do the opposite of the sine, cosine, and tangent functions. We write them using the superscript negative one, and we read them as the inverse sine, inverse cosine, and inverse tangent functions. They’re also known as the arc sine, arc cosine, and arc tangent functions.

It is important that we realize that this superscript of negative one does not mean the reciprocal. The inverse sin of 𝑥 is not the same as one over sin 𝑥. These inverse trigonometric functions are another way of describing the relationship between an angle and the values of its three trigonometric ratios. Their interpretation is as follows. If there is a value 𝑥 such that 𝑥 is equal to sin of 𝜃, then we can equivalently write this as 𝜃 is equal to the inverse sin of 𝑥. In the same way, if there is a value 𝑦 such that 𝑦 is equal to the cos of 𝜃, then 𝜃 is equal to the inverse cos of 𝑦. And if there is a value 𝑧 such that 𝑧 is equal to the tan of 𝜃, then 𝜃 is equal to the inverse tan of 𝑧.

If we know the value of one of the three trigonometric ratios, so the value of either 𝑥, 𝑦, or 𝑧, we can work backward to find the angle associated with this ratio by applying an inverse trigonometric function. To find these functions on our calculators, we usually have to press shift and then either the sin, cos, or tan button to get the inverse of each function. We’ll begin by looking at an example of how we can use these functions to find the measure of an angle given two side lengths in a right triangle.

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

Let’s begin by identifying angle 𝐵𝐴𝐶 on the diagram. It’s the angle formed when we travel from 𝐵 to 𝐴 to 𝐶, so it’s this angle here. We can denote this using the Greek letter 𝜃 if we wish. We see that the triangle we’re given is a right triangle, in which we know the lengths of two of the sides, and we wish to calculate the measure of one angle. We can therefore approach this problem using right triangle trigonometry.

We’ll begin by labeling the three sides of the triangle in relation to this angle 𝜃. The longest side of the triangle, which is the side directly opposite the right angle, is the hypotenuse. The side opposite this angle 𝜃, that’s side 𝐵𝐶, is the opposite. And the side between angle 𝜃 and the right angle, side 𝐴𝐵, is the adjacent. To help us decide which of the three trigonometric ratios we need in this question, we can recall the acronym SOH CAH TOA. The two side lengths we’re given for this triangle are the opposite and the adjacent. So this tells us it is the tangent ratio we need to use.

For an angle 𝜃 in a right triangle, the tan of angle 𝜃 is defined to be equal to the length of the opposite side divided by the length of the adjacent. So for this triangle, we have that the tangent of angle 𝜃 is equal to seven over five. To find the value of 𝜃, we need to apply the inverse tangent function. We have that 𝜃 is equal to the inverse tan of seven over five. We can evaluate this on our calculators. We usually need to press shift and then the tan function to bring up the inverse tangent function.

We’ve been asked to give our answer in degrees, so we must also make sure that our calculators are in degree mode. Evaluating the inverse tan of seven over five gives 54.4623 continuing. The question specifies that we should give our answer to two decimal places. And as the digit in the third decimal place is a two, we round down to 54.46. By applying the inverse tangent function then, we found that the measure of angle 𝐵𝐴𝐶 in degrees to two decimal places is 54.46 degrees.

In our next example, we’ll find both of the unknown angles in a right triangle using two different inverse trigonometric functions.

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

Looking at the digram, we see that we have a right triangle in which we know the lengths of two sides. We can therefore approach this problem using right triangle 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 𝐴𝐵𝐶 first, and we can label this angle as 𝜃. The side directly opposite this angle is the opposite. The side between this angle and the right angle is the adjacent. And the final side, which is always directly opposite the right angle is the hypotenuse.

To decide which of the three trigonometric ratios we need to use, we can recall the acronym SOH CAH TOA. In relation to angle 𝐴𝐵𝐶 or angle 𝜃, the two sides whose length we’ve been given are the adjacent and the hypotenuse. So it is the cosine ratio that we need to use. For an angle 𝜃 in a right triangle, the cos of angle 𝜃 is defined to be equal to the length of the adjacent side divided by the length of the hypotenuse. So substituting the lengths we know, we have the cos of 𝜃 is equal to four-ninths. To find the value of 𝜃, we need to apply the inverse cosine function. This is the function that says if cos of 𝜃 is four-ninths, what is the value of 𝜃?

We can evaluate this on our calculators, usually by pressing shift and then the cos button to bring up the inverse cosine function. And it gives 63.6122 continuing. The question specifies that we should give our answer to two decimal places, so we round to 63.61 degrees. Next, we need to calculate the measure of angle 𝐴𝐶𝐵, which we can label on our diagram as angle 𝛼. Now, we could calculate this angle, using the fact that angles in a triangle sum to 180 degrees. But instead, we’ll calculate this angle using trigonometry and then check our answer by summing the three angles.

Now importantly, because we’re calculating a different angle, we need to relabel the sides in the triangle. The hypotenuse of a right triangle is always the same side. It’s the side directly opposite the right angle. But the adjacent and the opposite sides are labeled in relation to the angle we’re calculating. The opposite is the side directly opposite this angle. So in relation to angle 𝛼, it’s the side 𝐴𝐵, which is the opposite. And then in relation to angle 𝛼, it’s the side 𝐴𝐶, which is the adjacent.

We now see that in relation to angle 𝛼, it is the opposite and the hypotenuse whose lengths we know. And so this time, we’re going to need to use the sine ratio. For an angle 𝛼 in a right triangle, sin of 𝛼 is defined to be equal to the length of the opposite divided by the length of the hypotenuse. So this time, we have that sin of 𝛼 is equal to four-ninths. To find the value of 𝛼, we need to apply the inverse sine function, giving 𝛼 is equal to the inverse sine of four-ninths. Evaluating this on a calculator, which must be in degree mode, gives 26.3877 continuing, and this rounds to 26.39 to two decimal places.

We can check our answer by summing the measures of the three angles in the triangle and confirming that this is indeed equal to 180 degrees. So by applying two different trigonometric ratios and then their inverse trigonometric functions, we found the measure of angle 𝐴𝐵𝐶 is 63.61 degrees and the measure of angle 𝐴𝐶𝐵 is 26.39 degrees, each to two decimal places.

In the examples we’ve seen so far, we found either one or two missing angles in a right triangle using the inverse trigonometric functions. Sometimes we may be required to go further than this and find all the missing angles and all the unknown side lengths in a right triangle. This is called solving a triangle. And we’ll practice this in our next example.

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.

Let’s now summarize the key points from this video. When working with right triangles, we use the terms opposite, adjacent, and hypotenuse to refer to the sides of the triangle. The hypotenuse is always opposite the right angle and is the longest side. The opposite and the adjacent are labeled in relation to a given angle, often denoted 𝜃. The opposite is the side directly opposite this angle, and the adjacent is the side between this angle and the right angle.

The acronym SOH CAH TOA can help us remember the definitions of the three trigonometric ratios. sin of 𝜃 is equal to the opposite over the hypotenuse, cos of 𝜃 is equal to the adjacent over the hypotenuse, and tan of 𝜃 is equal to the opposite over the adjacent. We can find the measure of an angle in a right triangle given two side lengths using the inverse trigonometric functions. If there is a value 𝑥 such that 𝑥 is equal to sin 𝜃, then 𝜃 is equal to the inverse sin of 𝑥. If 𝑦 is equal to cos 𝜃, then 𝜃 is equal to the inverse cos of 𝑦. And if 𝑧 is equal to tan of 𝜃, then 𝜃 is equal to the inverse tan of 𝑧.

We saw that we can use the three trigonometric functions and their inverses to solve triangles, which means to find the length of all unknown sides and the measures of all unknown angles. When doing this, we may also use the Pythagorean theorem or the angle sum in a triangle, either as an alternative method or to check our answers.