In this explainer, we will learn how to find a missing angle in a right triangle using the appropriate inverse trigonometric function given two side lengths.
When working with right triangle trigonometry, it is useful to recall the acronym “SOH CAH TOA.” This helps us remember the definitions of the trigonometric ratios sine, cosine, and tangent in terms of the sides relative to an angle which we call the opposite, adjacent, or hypotenuse. Let us list the ratios here.
The hypotenuse is always the longest side of the right triangle, the opposite side is the side directly opposite the angle concerned, and the adjacent is the side next to the angle (which is not the hypotenuse). An example of this is shown here.
In order to find the measures of unknown angles in right triangles (using trigonometry) we need to be confident in our ability to correctly label the triangle in terms of the opposite, adjacent, and hypotenuse and correctly remember the trigonometric ratios. Once we are happy with these two things, we are in a position to start solving trigonometry problems involving finding the measure of an unknown angle.
Let us start by looking at an example.
Example 1: Finding the Measure of an Unknown Angle in a Right Triangle
For the given figure, find the measure of angle , in degrees, to two decimal places.
Our first step in answering this question is to label the triangle relative to the angle .
Notice here that we have circled A and H as these are the two sides whose lengths we know. If we then recall the acronym “SOH CAH TOA,” we can see that “CAH” is the only one that contains A and H, which means that we need to use the cosine ratio. Recall that
We now substitute the values of A and H to find that
By using the properties of inverse cosine, we find that
If we then calculate this, we get
In some questions, we may be asked to calculate the measures of all the unknown angles in a right triangle. In this case, we have to use trigonometry to find one of the unknown angles and then we can use the fact that the measures of the angles in a triangle sum to . Let us look at an example where this is the case.
Example 2: Finding the Measures of all the Unknown Angles in a Right Triangle
For the given figure, find the measures of and , in degrees, to two decimal places.
Our first step is to choose one of the two unknown angles to calculate first. Here, we are going to start by finding which we will call . We can then label the sides of the triangle in relation to the angle as shown.
We have circled O and A as these are the lengths that we know. If we then recall the acronym “SOH CAH TOA,” we can see that we need to use the tangent ratio as “TOA” contains the letters O and A. Recall that
Substituting the lengths O and A, we get
Using the properties of inverse tangent, we find that
If we calculate this, we find that
To find the measure of the second unknown angle in the triangle, we need to use the fact that the sum of the measures of the angles in a triangle is . If we call , we have that
This simplifies to and subtracting 128.66 from both sides, we find that
In some trigonometry questions, we are not given a diagram and part of the skill of the question is drawing an appropriate diagram. In the following example, we will demonstrate this skill.
Example 3: Solving Triangles with Trigonometry
is a right triangle at where cm and . Find the length to the nearest centimeter and the measures of angles and to the nearest degree.
We begin by making a diagram. It is usually helpful to try and draw approximately to scale. It is not completely necessary, but it does help us check that our answers are reasonable when comparing them to the diagram. We, therefore, draw a triangle and label the edge lengths we know.
The first thing we have been asked to find is the length . To do this, we can use the Pythagorean theorem which states that where is the length of the hypotenuse. In the triangle we have been given, is the hypotenuse. Hence, we can write the Pythagorean theorem for the triangle as
Substituting in and , we have
Taking the square root, we have to the nearest centimeter.
We now need to find the measures of the angles at and . To do this, we can find one of the angles and then use the fact that the angles in a triangle sum to . We will find the measure of which we will denote . To know which trigonometric ratio we should use, we first need to label the sides of the triangle. We know that is the hypotenuse. Since we are considering , is the opposite and is the adjacent.
Since we know the lengths of all the sides, we could use any trigonometric ratio. However, it is best to use the two lengths we were given in the question. There are two good reasons for this. First, this means that if we made a mistake calculating the third side, it will not affect our answer to this part of the question. Second, we could easily make rounding errors if we use the length of the third side since its exact form is not a whole number. Therefore, we would like to calculate using the opposite and the hypotenuse. This means that we will use the sine ratio:
Substituting in the lengths of the opposite () and the hypotenuse (), we have
Using the inverse sine, we have
Using a calculator, we can evaluate this and find to the nearest degree. Therefore, to the nearest degree.
We can now use the fact that the angles in a triangle sum to to find . Since we have
Substituting in the values of and , we have to the nearest degree.
Trigonometry questions may also be presented as story problems. When this is the case, if an associated diagram is not given, it is always worth drawing one. An example of this type of question would be the following:
Example 4: Solving Story Problems with Trigonometry
A 5 m ladder is leaned against a perpendicular wall such that its base is 2 m from the wall. Work out the angle between the ladder and the floor, giving your answer to two decimal places.
Our first step in solving a question like this is to draw a diagram of the situation.
In this diagram, relative to the angle , we have labeled the sides whose lengths we know. Here we know the length of the adjacent and the hypotenuse so we need to use the cosine ratio to find the measure of the unknown angle. We know that
If we substitute in the lengths A and H, we get
If we then use the properties of inverse cosine, we find that
Calculating this, we find that
We will finish by considering one final story problem.
Example 5: Solving Story Problems with Trigonometry
The height of a ski slope is 16 meters and the length is 20 meters. Find the measure of giving the answer to two decimal places.
In this question, we are fortunate to have been given an associated diagram which means we do not need to draw this ourselves. Our first step is to label the sides relative to the angle theta.
Here, we know the lengths of the opposite and the hypotenuse and, therefore, need to use the sine ratio to find the measure of the unknown angle. Recall that If we substitute in the lengths O and H, we get
If we then use the properties of inverse sine, we find that
Calculating this, we find that
- 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 . The adjacent is the side next to the angle which is not the hypotenuse. As for the opposite, it is the last side of the triangle. It is called the opposite since it is opposite the given angle.
- Recall the acronym “SOH CAH TOA,” where O stands for the opposite, A stands for the adjacent, H stands for the hypotenuse, and is the angle. The trigonometric ratios are
- We can find the measure of an angle given the side lengths using inverse trigonometric functions.