Explainer: Applications of Inverse Trigonometric Functions in a Right Triangle

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 β€œSOH 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.

Trigonometric Ratios

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 πœƒ, in degrees, to two decimal places.

Answer

Our first step in answering this question is to label the triangle relative to the angle πœƒ.

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 β€œSOH CAH TOA,” we can see that β€œCAH” is the only one that contains A and H, which means that we need to use the cosine ratio. Recall that cosAHπœƒ=.

We now substitute the values of A and H to find that cosπœƒ=38.

By using the properties of inverse cosine, we find that πœƒ=ο€Ό38.cos

If we then calculate this, we get 67.98(2).∘d.p.

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 180∘. 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 ∠𝐴𝐢𝐡 and ∠𝐡𝐴𝐢, in degrees, to two decimal places.

Answer

Our first step is to choose one of the two unknown angles to calculate first. Here, we are going to start by finding ∠𝐴𝐢𝐡 which we will call π‘₯. We can then label the sides of the triangle in relation to the angle π‘₯ as shown.

We have circled O and A as these are the lengths that we know. If we then recall the acronym β€œSOH CAH TOA,” we can see that we need to use the tangent ratio as β€œTOA” contains the letters O and A. Recall that tanOAπ‘₯=.

Substituting the lengths O and A, we get tanπ‘₯=45.

Using the properties of inverse tangent, we find that π‘₯=ο€Ό45.tan

If we calculate this, we find that π‘₯=38.66.∘

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 180∘. If we call βˆ π΅π΄πΆπ‘¦, we have that 𝑦+38.66+90=180.

This simplifies to 𝑦+128.66=180, and subtracting 128.66 from both sides, we find that 𝑦=51.34.∘

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

𝐴𝐡𝐢 is a right triangle at 𝐡 where 𝐡𝐢=10 cm and 𝐴𝐢=18cm. Find the length 𝐴𝐡 to the nearest centimeter and the measures of angles 𝐴 and 𝐢 to the nearest degree.

Answer

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 𝐴𝐡𝐢 and label the edge lengths we know.

The first thing we have been asked to find is the length 𝐴𝐡. To do this, we can use the Pythagorean theorem which states that 𝑐=π‘Ž+𝑏, where 𝑐 is the length of the hypotenuse. In the triangle we have been given, 𝐴𝐢 is the hypotenuse. Hence, we can write the Pythagorean theorem for the triangle as 𝐴𝐢=𝐴𝐡+𝐡𝐢.

Therefore, 𝐴𝐡=π΄πΆβˆ’π΅πΆ.

Substituting in 𝐡𝐢=10 and 𝐴𝐢=18, we have 𝐴𝐡=18βˆ’10=324βˆ’100=224.

Taking the square root, we have 𝐴𝐡=√224=14.966…=15cm to the nearest centimeter.

We now need to find the measures of the angles at 𝐴 and 𝐢. To do this, we can find one of the angles and then use the fact that the angles in a triangle sum to 180∘. We will find the measure of ∠𝐴 which we will denote πœƒ. To know which trigonometric ratio we should use, we first need to label the sides of the triangle. We know that 𝐴𝐢 is the hypotenuse. Since we are considering ∠𝐴, 𝐡𝐢 is the opposite and 𝐴𝐡 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 ∠𝐴 using the opposite and the hypotenuse. This means that we will use the sine ratio: sinOHπœƒ=.

Substituting in the lengths of the opposite (𝐡𝐢=10) and the hypotenuse (𝐴𝐢=18), we have sinπœƒ=1018=59.

Using the inverse sine, we have πœƒ=ο€Ό59.sin

Using a calculator, we can evaluate this and find πœƒ=33.748…=34∘ to the nearest degree. Therefore, π‘šβˆ π΄=34∘ to the nearest degree.

We can now use the fact that the angles in a triangle sum to 180∘ to find π‘šβˆ πΆ. Since π‘šβˆ π΄+π‘šβˆ π΅+π‘šβˆ πΆ=180, we have π‘šβˆ πΆ=180βˆ’π‘šβˆ π΅βˆ’π‘šβˆ π΄.

Substituting in the values of π‘šβˆ π΅ and π‘šβˆ πΆ, we have π‘šβˆ πΆ=180βˆ’90βˆ’33.748…=56.251…=56∘ 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.

Answer

Our first step in solving a question like this is to draw a diagram of the situation.

In this diagram, relative to the angle π‘₯, 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 cosAHπ‘₯=.

If we substitute in the lengths A and H, we get cosπ‘₯=25.

If we then use the properties of inverse cosine, we find that π‘₯=ο€Ό25.cos

Calculating this, we find that π‘₯=66.42.∘

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 βˆ πœƒ giving the answer to two decimal places.

Answer

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 sinOHπœƒ=. If we substitute in the lengths O and H, we get sinπœƒ=1620.

If we then use the properties of inverse sine, we find that πœƒ=ο€Ό1620.sin

Calculating this, we find that πœƒ=53.13.∘

Key Points

  1. 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 πœƒ. The adjacent is the side next to the angle πœƒ 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.
  2. Recall the acronym β€œSOH CAH TOA,” where O stands for the opposite, A stands for the adjacent, H stands for the hypotenuse, and πœƒ is the angle. The trigonometric ratios are sinOHcosAHandtanOAπœƒ=,πœƒ=,πœƒ=.
  3. We can find the measure of an angle given the side lengths using inverse trigonometric functions.

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