Explainer: Right Triangle Trigonometry

In this explainer, we will learn how to find the value of a missing side length in a right triangle by choosing the appropriate trigonometric ratio for a given angle.

Recall, when working with right triangle trigonometry, it is useful to recall the acronym β€œSOH CAH TOA.” This helps us to remember the definitions of the trigonometric ratiosβ€”sine, cosine, and tangentβ€”in terms of the sides relative to an angle which we call the opposite, adjacent, and hypotenuse. Let us list the ratios here.

Trigonometric Ratios

The hypotenuse is always the longest side of the right triangle (directly opposite the right angle), 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.

When we are confident that we can remember the three trigonometric ratios and correctly label the sides of a right triangle, we can start looking at how to calculate unknown lengths of a right triangle. When calculating these lengths, we can categorize them into two distinct types of questions. Once you have labeled the right triangle and substituted the values into the correct trigonometric ratio, you will find with some questions the unknown appears at the top of the fraction, and sometimes it appears at the bottom. We will look at detailed examples of both cases. A very common mistake is to assume that the unknown always appears on the top of the fraction: this is a mistake that is made due to not labeling the triangle correctly.

Let us start by looking at an example where the unknown appears at the top of the fraction.

Example 1: Finding Unknown Lengths in a Right Triangle Where the Unknown Is at the Top of the Fraction

Find π‘₯ to two decimal places.

Answer

Our first step when solving any problem involving finding lengths of right triangles is to label the sides relative to the known angle, in this case the angle with measure 55∘.

We can note here that we did not need to label the adjacent side as we neither know its length nor are we trying to find it. The sides that we are interested in are the opposite and the hypotenuse, which, if you recall the three trigonometric ratios, means that we need to use sine. Recall that sinOHπœƒ=.

If we then substitute our values for O, H, and πœƒ, we get sin55=π‘₯10.∘

To solve this, we multiply both sides by 10 to get π‘₯=10Γ—55.sin∘

Calculating this, we find that π‘₯=8.19.(2d.p.)

Let us look at a second example of this where the triangle is described by its vertices.

Example 2: Finding Unknown Lengths in a Right Triangle Where the Unknown Is at the Top of the Fraction

Find the length of 𝐡𝐢 giving the answer to two decimal places.

Answer

Our first step when solving any problem involving finding lengths of right triangles is to label the sides relative to the known angle, in this case ∠𝐡𝐴𝐢. At this point it is also helpful if we refer to the length 𝐡𝐢 as π‘₯.

We can note here that we did not need to label the adjacent side as we neither know its length nor are we trying to find it. The sides that we are interested in are the opposite and the hypotenuse, which, if you recall the three trigonometric ratios, means that we need to use sine. Recall that sinOHπœƒ=.

If we then substitute our values for O, H, and πœƒ, we get sin47=π‘₯15.∘

To solve this, we multiply both sides by 15 to get π‘₯=15Γ—47.sin∘

Calculating this, we find that π‘₯=10.97.(2d.p.)

Now, let us move on to looking at examples of questions where the unknown appears on the bottom of the fraction. With these questions, we have an additional step in our working out, so a little bit of care needs to be taken in our calculations.

Example 3: Finding Unknown Lengths in a Right Triangle Where the Unknown Is at the Bottom of the Fraction

Find π‘₯ to two decimal places.

Answer

Our first step when solving any problem involving finding lengths of right triangles is to label the sides relative to the known angle, in this case the angle with measure 20∘.

We can note here that we did not need to label the adjacent side as we neither know its length nor are we trying to find it. The sides that we are interested in are the opposite and the hypotenuse, which, if you recall the three trigonometric ratios, means that we need to use sine. Recall that sinOHπœƒ=.

If we then substitute our values for O, H, and πœƒ, we get sin20=12π‘₯.∘

This is slightly more difficult to solve as we need to multiply both sides by π‘₯ first to get π‘₯Γ—20=12,sin∘ and then dividing each side by sin20∘, we find that π‘₯=1220.sin∘

Calculating this, we find that π‘₯=35.09.(2d.p.)

Now, let us have a look at a couple of questions that are story problems. These contain the additional step of having to draw an associated diagram, being careful to interpret the information from the question correctly.

Example 4: Solving Word Problems with Trigonometry

A person observes a point on the ground from the top of a hill that is 1.56 km high. The angle of depression is 29∘. Find the distance between the point and the observer giving the answer to the nearest meter.

Answer

The first thing we should do when presented with a word problem in trigonometry is to sketch the triangle described in the problem marking all the angles and lengths we know. Before we can do this, it is important to understand what we mean when we talk about an angle of depression. This represents the angle below a horizontal. Therefore, to mark this angle in our figure, we need to draw a horizontal from the observer, 𝑂. Then, the line from the observer to the point 𝑃 on the ground will need to make an angle of 29∘ with this horizontal.

Considering the triangle 𝑂𝑃𝐡, we can find the measure of angle βˆ π΅π‘‚π‘ƒ by taking 29∘ from 90∘. Hence, π‘šβˆ π΅π‘‚π‘ƒ=90βˆ’29=61.∘

We can now use trigonometry to find the distance between the observer and the point. This is given by the length 𝑂𝑃. To ensure we use the correct trigonometric identity, we need to correctly label the sides in our triangle. The hypotenuse is 𝑂𝑃 since this is opposite the right angle. Since we would like to label the sides relative to the known angle, we see that 𝑂𝐡 is the adjacent side.

Hence, we would like to find the hypotenuse where we know the adjacent side. The trigonometric ratio that links these sides is the cosine ratio. In particular, cosAHπœƒ==𝑂𝐡𝑂𝑃.

Since we would like to calculate the length 𝑂𝑃, we can make this the subject of the equation by multiplying both sides by 𝑂𝑃 as follows: π‘‚π‘ƒπœƒ=𝑂𝐡.cos

We can now divide both sides by cosπœƒ to isolate 𝑂𝑃 on the left-hand side as follows: 𝑂𝑃=π‘‚π΅πœƒ.cos

Substituting in 𝑂𝐡=1.56 and πœƒ=61, we have 𝑂𝑃=1.5661.cos∘

Using a calculator, we can evaluate the expression on the right-hand side to find 𝑂𝑃=3.217757….km

Since we have been asked to give the answer to the nearest meter, we should multiply by 1,000 and then round as follows: 𝑂𝑃=32,177.57…=3,218m to the nearest meter.

Example 5: Finding Unknown Lengths in a Right Triangle Where the Unknown Is at the Top of the Fraction

A person is trying to estimate the height of the Eiffel Tower. He measured a distance of 250 m from the base of the tower. From that point, he measured the angle of elevation to the top of the tower to be 52∘. Use these measurements to approximate the height of the tower to the nearest meter.

Answer

We begin by drawing a diagram to represent the situation labeling the edges of the triangle as opposite, adjacent, and hypotenuse.

As we can see, the unknown height represents the side opposite the angle and we know the adjacent side. We, therefore, need the trigonometric ratio that links the opposite and adjacent sides, which is the tangent ratio: tanOAπœƒ=.

We would like to find the height represented by the side opposite the angle. Therefore, by multiplying by A we can make O the subject of the equation as follows: OAtan=πœƒ.

Substituting in the length of the adjacent side is 250 and the angle πœƒ=52∘, we have Otanm=25052=319.98…=320∘ to the nearest meter. Hence, according to the calculations, the height of the Eiffel tower is 320 meters to the nearest meter.

Example 6: Finding Unknown Lengths in a Right Triangle Where the Unknown Is at the Top of the Fraction

A 23 ft ladder leans against a building such that the angle between the ground and the ladder is 80∘. How high does the ladder reach up the side of the building? Give your answer to two decimal places.

Answer

Our first step when solving this problem is to draw a diagram, labeling the opposite, adjacent, and hypotenuse in the process.

The sides that we are interested in are the opposite and the hypotenuse, which, if you recall the three trigonometric ratios, means that we need to use sine. Recall that sinOHπœƒ=.

If we then substitute our values for O, H, and πœƒ, we get sin80=π‘₯23.∘

To solve this, we multiply both sides by 23 to get π‘₯=23Γ—80.sin∘

Calculating this, we find that π‘₯=22.65.(2d.p.)

Example 7: Finding Unknown Lengths in a Right Triangle Where the Unknown Is at the Top of the Fraction

A kite, which is at a perpendicular height of 44 m, is attached to a string inclined at 60∘ to the horizontal. Find the length of the string accurate to one decimal place.

Answer

Our first step when solving this problem is to draw a diagram, labeling the opposite, adjacent, and hypotenuse in the process.

The sides that we are interested in are the opposite and the hypotenuse, which, if you recall the three trigonometric ratios, means that we need to use sine. Recall that sinOHπœƒ=.

If we then substitute our values for O, H, and πœƒ, we get sin60=44π‘₯.∘

To solve this, we start by multiplying both sides by π‘₯ to get 44=π‘₯Γ—60,sin∘ and then dividing each side by sin60∘, we find that π‘₯=4460.sin∘

Calculating this, we find that π‘₯=50.8.(1d.p.)

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 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. To find a missing side length, we follow the following set of steps:
    1. Label the sides in the triangle as opposite, adjacent, and hypotenuse relative to the known angle.
    2. Choose the correct trigonometric ratio which links the known side to the unknown side.
    3. Rearrange the formula for the ratio to make the unknown side the subject.
    4. Substitute in the values of the known side and angle.

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