Lesson Explainer: Right Triangle Trigonometry: Solving for a Side Mathematics • 11th Grade

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 that a trigonometric ratio is a ratio of the lengths of two different sides of a right triangle and that the group of all trigonometric ratios is the right triangle trigonometry. When working with right triangle trigonometry, it is useful to recall the acronyms “SOH CAH TOA.” This helps us to 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, and hypotenuse. Let us list the ratios here.

Definition: 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).

When we are given the length of one side of the right triangle and an angle 𝜃 in the right triangle, different from the right angle, we can use right triangle trigonometry to compute the lengths of the remaining sides. We have three different choices for trigonometric ratios: sine, cosine, and tangent. Depending on the given information, an angle and a side, we will need to decide which trigonometric ratio to use. For instance, consider a situation where we need to find the length of the hypotenuse when given an angle 𝜃 and the length of the opposite side of the right triangle; In this case, we will need to use the trigonometric ratio that relates the hypotenuse and opposite sides of the right triangle, which is sine.

When we perform these calculations, we often need to use calculators to find the values of trigonometric ratios. But, if a calculator’s unit for angle is set in radians when we are using an angle given in degrees, the calculator will give an incorrect value for the trigonometric ratio. Most scientific calculators capable of computing trigonometric ratios will have a default setting for angular units and also an option to switch between radians and degrees. We should always make sure that our calculator is set to the same unit as what is used in each problem.

In our first example, we will find the length of the opposite side when given an angle and the length of the hypotenuse of a right triangle.

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

Find 𝑥 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 we recall the three trigonometric ratios, means that we need to use sine. Recall that sinOH𝜃=.

If we then substitute our values for O, H, and 𝜃, we get the following: sin55=𝑥10.

To solve this, we multiply both sides by 10 to get the following: 𝑥=10×55.sin

Evaluating this value using a calculator and making sure that the unit of angle is set to degrees, we find that 𝑥=8.19(2,).d.p.roundedtotwodecimalplaces

In the previous example, we found an unknown length of a side of a right triangle when we were given an angle (different from the right angle) and the length of another side of the right triangle. Note that we began by labeling each side of the right triangle as O (Opposite), A (Adjacent), and H (Hypotenuse), and then recalling the suitable trigonometric ratio. We summarize these general steps of identifying a missing length by using right triangle trigonometry.

How To: Finding a Missing Side Length using Right Triangle Trigonometry

When given an angle (different from the right angle) and the length of a side of a right triangle, we can find the length of another side of the right triangle by following these steps:

  1. Label the sides in the triangle as opposite, adjacent, and hypotenuse in accordance with the known angle.
  2. Choose the correct trigonometric ratio that 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.

Let us look at a second example of this where we can apply this method to find a missing side length in a right triangle.

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

Find the length of 𝐵𝐶 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 𝐵𝐴𝐶. At this point it is also helpful if we refer to the length 𝐵𝐶 as 𝑥.

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 we recall the three trigonometric ratios, means that we need to use sine. Recall that sinOH𝜃=.

If we then substitute our values for O, H, and 𝜃, we get sin47=𝑥15.

To solve this, we multiply both sides by 15 to get 𝑥=15×47.sin

Evaluating this value using a calculator and making sure that the unit of angle is set to degrees, we find that 𝑥=10.97(2,).d.p.roundedtotwodecimalplaces

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 𝑥 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 we recall the three trigonometric ratios, means that we need to use sine. Recall that sinOH𝜃=.

If we then substitute our values for O, H, and 𝜃, we get sin20=12𝑥.

We multiply both sides by 𝑥 to get 𝑥×20=12.sin Then, we divide each side by sin20 and get 𝑥=1220.sin

Evaluating this value using a calculator and making sure that the unit of angle is set to degrees, we find that 𝑥=35.09(2,).d.p.roundedtotwodecimalplaces

Now, let us have a look at a couple of questions that are real-world 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: 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 we recall the three trigonometric ratios, means that we need to use sine. Recall that sinOH𝜃=.

If we then substitute our values for O, H, and 𝜃, we get sin80=𝑥23.

To solve this, we multiply both sides by 23 to write 𝑥=23×80.sin

Evaluating this value using a calculator and making sure that the unit of angle is set to degrees, we find that 𝑥=22.65(2,).d.p.roundedtotwodecimalplaces

Example 5: 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 we recall the three trigonometric ratios, means that we need to use sine. Recall that sinOH𝜃=.

If we then substitute our values for O, H, and 𝜃, we get sin60=44𝑥.

To solve this, we start by multiplying both sides by 𝑥 to get 44=𝑥×60,sin and then dividing each side by sin60, we find that 𝑥=4460.sin

Evaluating this value using a calculator and making sure that the unit of angle is set to degrees, we find that 𝑥=50.8(1,).d.p.roundedtoonedecimalplace

Let us finish by recalling a few important concepts from this explainer.

Key Points

  • When working with right triangles, we use the terms opposite, adjacent, and hypotenuse to refer to the sides of the triangle.
  • Recall the acronyms “SOH CAH TOA,” where O stands for the opposite, A stands for the adjacent, H stands for the hypotenuse, and 𝜃 is the angle. The trigonometric ratios are sinOHcosAHandtanOA𝜃=,𝜃=,𝜃=.
  • When given an angle (different from the right angle) and the length of a side of a right triangle, we can find the length of another side of the right triangle by following these steps:
    • Label the sides in the triangle as opposite, adjacent, and hypotenuse in accordance with the known angle.
    • Choose the correct trigonometric ratio that links the known side to the unknown side.
    • Rearrange the formula for the ratio to make the unknown side the subject.
    • Substitute in the values of the known side and angle.

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