Lesson Video: Right Triangle Trigonometry: Solving for an Angle Mathematics • Third Year of Preparatory School

Video Transcript

In this video, we will learn how to find a 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 like the one shown, with one of the non-right angles labeled as 𝜃. The hypotenuse of a right triangle is its longest side, which is always the side directly opposite the right angle. In relation to the angle we’ve labeled as 𝜃, the side directly opposite this angle is called the opposite. And the side between the right angle and the angle 𝜃 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 the different pairs of these side lengths. 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 SOHCAHTOA to help us remember the definitions of the three trigonometric ratios. The first letter in each part refers to the trigonometric ratio we’re using, either sine, cosine, or tangent. And then the next two letters refer to the numerator and denominator of the sides involved in the ratio. So SOH tells us that sin of an angle 𝜃 is equal to the length of the opposite side divided by the length of the hypotenuse, O over H. cos of 𝜃 is equal to the adjacent over the hypotenuse. And tan of 𝜃 is equal to the opposite over 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 lengths of two of the triangle’s side. To do this, we’ll need to use the inverse trigonometric functions. These are essentially the functions that do the opposite of what the sine, cosine, and tangent functions do. We denote these using the superscript negative one. And we read this as the inverse sine, inverse cosine, and inverse tangent functions. They are also known as the arcsine, arccosine, and arctangent functions.

These inverse trigonometric functions are an alternative way of describing the relationship between an angle and the values of its three trigonometric ratios. We interpret them as follows. If there is a value 𝑥 such that 𝑥 is equal to sin 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 𝑧 is equal to the tan of 𝜃, then 𝜃 is equal to the inverse tan of 𝑧.

Now, it’s important to note that this notation does not mean a reciprocal. The inverse sin of 𝑥 does not mean one over sin of 𝑥. What this means is that if we know the value of one of the three trigonometric ratios for an angle 𝜃, we can work backward to find the angle associated with this ratio by applying the inverse trigonometric function. To access 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.

Let’s look at an example of how we can use these inverse 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 𝜃 in degrees to two decimal places.

Looking at the figure, we can see that we have a right triangle, in which 𝜃 represents the measure of one of the non-right angles. We’ve also been given the lengths of two of the sides of the right triangle. They are three and eight units. We can therefore approach this problem using trigonometry.

Our first step in any problem involving trigonometry is to label the three sides of the triangle in relation to the angle 𝜃. The side directly opposite the right angle is the hypotenuse, which we’ll abbreviate to H. The side directly opposite the angle 𝜃 is called the opposite, abbreviated to O. And the side between the angle 𝜃 and the right angle is the adjacent, which we’ll abbreviate to A.

We’ll now recall the acronym SOHCAHTOA to help us decide which of the three trigonometric ratios we need to use in this problem. The two side lengths we’ve been given are the adjacent and the hypotenuse. So we’re going to be using the cosine ratio. The cos of an angle 𝜃 is defined to be the length of the adjacent side divided by the length of the hypotenuse. Substituting the values for this triangle, we have that cos of 𝜃 is equal to three over eight, or three-eighths.

Now we need to find the value of 𝜃, which means we need to apply the inverse cosine function. This is the function that essentially does the opposite of the cosine function. It says if cos of 𝜃 is equal to three-eighths, then what is 𝜃? We have 𝜃 is equal to the inverse cos of three-eighths.

We can then evaluate this on our calculators, making sure they’re in degree mode. To access the inverse cosine function, we usually need to press shift and then the cos button on our calculators. Evaluating gives 67.975. And then we round to two decimal places as specified in the question. So, by applying the inverse cosine function in this right triangle, we found that the measure of angle 𝜃 to two decimal places is 67.98 degrees.

Let’s now consider another example in which we’ll find the measures of two angles in a right triangle.

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.

In the two problems we’ve considered so far, we’ve been given a diagram of the right triangle we’re working with. However, in some trigonometry questions, we won’t be given a diagram. And part of the skill of answering the question is drawing an appropriate diagram ourselves. Let’s now consider an example of this.

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 measures of angles 𝐴 and 𝐶, giving the answer to the nearest degree.

We haven’t been given a diagram for this problem. So we need to begin by drawing one ourselves. We’re told that 𝐴𝐵𝐶 is a right triangle at 𝐵, which means it’s angle 𝐵 that is the right angle. We’re also told that 𝐵𝐶 is 10 centimeters and 𝐴𝐶 is 18 centimeters. We were asked to find the length 𝐴𝐵 and the measures of each of the other two angles in this triangle.

Let’s begin with finding the length of 𝐴𝐵. As we have a right triangle in which we know two of its side lengths, we can apply the Pythagorean theorem to calculate the length of the third side. The Pythagorean theorem tells us that, in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. In our triangle, the two shorter sides are 𝐴𝐵 and 𝐵𝐶 and the hypotenuse is 𝐴𝐶. So we have the equation 𝐴𝐵 squared plus 𝐵𝐶 squared is equal to 𝐴𝐶 squared.

Substituting 10 for 𝐵𝐶 and 18 for 𝐴𝐶 gives 𝐴𝐵 squared plus 10 squared is equal to 18 squared. This simplifies to 𝐴𝐵 squared plus 100 equals 324. So 𝐴𝐵 squared is equal to 224. 𝐴𝐵 is then equal to the square root of 224, which is 14.9666, or 15 to the nearest integer. So we found the length of 𝐴𝐵. And now we need to consider calculating the measures of the two angles. Let’s start with angle 𝐴.

We’ll begin by labeling the three sides of the triangle in relation to this angle. The side directly opposite, that’s 𝐵𝐶, is the opposite. The side between this angle and the right angle is the adjacent. And the side directly opposite the right angle is the hypotenuse. We can then recall the acronym SOHCAHTOA to help us decide which trigonometric ratio to use to calculate this angle.

As we now know the lengths of all three sides in the triangle, we could use any of the three ratios. But it makes the most sense to use the two sides whose lengths we’re originally given in case we made any mistakes when we calculated the length of 𝐴𝐵. So we’re going to use the sine ratio. This is defined as sin of 𝜃 is equal to the opposite over the hypotenuse. Substituting 10 for the opposite and 18 for the hypotenuse and using 𝐴 to represent the angle at 𝐴, we have sin of 𝐴 is equal to 10 over 18. To calculate 𝐴, we need to apply the inverse sine function, giving 𝐴 equals the inverse sin of 10 over 18. Evaluating on our calculators, which must be in degree mode, we have 33.748, which to the nearest degree is 34.

Finally, we need to calculate the measure of the third angle in the triangle. As the angles in any triangle sum to 180 degrees, we can do this by subtracting the measures of the other two angles from 180 degrees, which gives 56 degrees. And so we’ve completed our solution. The length of 𝐴𝐵 to the nearest centimeter is 15 centimeters. The measure of angle 𝐴 and the measure of angle 𝐶, each to the nearest degree, are 34 and 56 degrees, respectively.

Problems involving trigonometry may also be presented as a story describing a practical situation. When this is the case, we may not be given a diagram. And part of the skill will be to produce one ourselves from the information given in the question. Let’s look at one final example of this.

A five-meter ladder is leaning against a vertical wall such that its base is two meters from the wall. Work out the angle between the ladder and the floor, giving your answer to two decimal places.

We haven’t been given a diagram to accompany this question. So the first step is going to be to draw one. We have a ladder leaning against a vertical wall. The triangle formed by the ladder, the floor, and the wall is a right triangle. And in fact, that’s all we really need to draw for our diagram. The ladder is five meters long, and its base is two meters away from the wall. We are asked to work out the angle between the ladder and the floor. So that’s this angle here, which we’ll call 𝑥.

We have a right triangle in which we know the lengths of two sides. And we want to calculate the measure of an angle. So we can apply trigonometry. We begin by labeling the three sides of the triangle in relation to the angle 𝑥. So we have the opposite, the adjacent, and the hypotenuse. We then recall the acronym SOHCAHTOA to help us decide which of the three trigonometric ratios to use in this problem. The side lengths we know are the adjacent and the hypotenuse. So we’re going to be using the cosine ratio. This is defined for an angle 𝜃 as the length of the adjacent divided by the length of the hypotenuse. Substituting two for the adjacent and five for the hypotenuse, we have that cos of 𝑥 is equal to two-fifths.

To find the value of 𝑥, we need to apply the inverse cosine function, giving 𝑥 is equal to the inverse cos of two-fifths. Evaluating this on our calculators, which must be in degree mode, gives 66.421. Finally, we round our answer to two decimal places. And we found that the angle between the ladder and the floor is 66.42 degrees.

Let’s now summarize the key points that we’ve seen in this video. First, we recalled the terminology associated with the three sides of a right triangle: the opposite, the adjacent, and the hypotenuse. We then recalled the definitions of the three trigonometric ratios — sine, cosine, and tangent — and the acronym SOHCAHTOA, which we can use to help remember their definitions.

We then saw that we can find the measure of an angle by applying the inverse trigonometric functions, which take the value of one of these three trigonometric ratios and tell us the angle associated with it. We saw that we can apply these techniques to a range of problems involving right triangles, including story problems, which describe a practical situation.