Lesson Explainer: Right Triangle Trigonometry: Solving for a Side | Nagwa Lesson Explainer: Right Triangle Trigonometry: Solving for a Side | Nagwa

Lesson Explainer: Right Triangle Trigonometry: Solving for a Side Mathematics

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.

The sine, cosine, and tangent ratios are three of the most fundamental tools in working with both right triangles and circles. To understand what these ratios are, we will first consider a few geometric properties that set up their definitions.

First, we recall that the measures of the internal angles in a triangle sum to 180. This means, if we have a right triangle with a nonright angle of 𝜃, then the remaining angle will have a measure of 90𝜃.

Second, we recall the AAA criterion for similarity that tells us that two triangles are similar if they have the same internal angles. If two triangles are similar, then they have the same shape and the corresponding sides are in proportion. In other words, the lengths of their sides are scalar multiples of each other.

These two facts allow us to notice an interesting property of right triangles: the value of the ratio of any two lengths of the sides of a right triangle is only dependent on the angle and the choice of the two sides. In other words, we can determine the value of the ratio of two sides of a right triangle from the measure of the angle.

To help us understand this, we first note that all right triangles are similar if they share one internal angle that is not the right angle. This is because, if two right triangles have internal angles of 𝜃 and 90, the third angle must have a measure of 90𝜃, so the triangles are similar by the AAA criterion. In particular, we recall that the ratios of corresponding sides in similar triangles are equal.

For example, since all right triangles with an internal angle of 60 are similar, let’s consider the following triangles, 𝐴𝐵𝐶 and 𝐴𝐵𝐶.

Since these triangles are similar, the ratio of their corresponding sides must be equal, for example 𝐵𝐶𝐵𝐶=𝑎𝑘𝑎=𝑘,𝐴𝐶𝐴𝐶=𝑏𝑘𝑏=𝑘.

Hence, 𝐵𝐶𝐵𝐶=𝐴𝐶𝐴𝐶.

We can then rearrange this equation to see that 𝐵𝐶𝐴𝐶=𝐵𝐶𝐴𝐶.

Therefore, all right triangles with an angle of 60 will have equal ratios of their own corresponding side lengths. Of course, there was nothing unique about the choice of 60; this result would be true for any angle we had chosen.

Before we state this result and discuss its uses, we need to clarify exactly what is meant by “corresponding sides.” In our above example, the corresponding sides were easy to determine since we could see which sides were scalar multiples of each other. However, in general, it is not this easy. So instead, we label each of the sides of the triangle based on their position relative to the angle.

Since there are three sides, we will need three labels for the sides of the triangle. First, we recall that the hypotenuse of a right triangle is its longest side and that it is always opposite the right angle. In triangle 𝐴𝐵𝐶, we see that 𝐴𝐵 is the side opposite the right angle, so this is the hypotenuse. Second, we can label the remaining sides by considering their positions relative to 𝐵. We see that 𝐴𝐶 is opposite the angle, so we will call this the opposite side, and 𝐵𝐶 is adjacent to the angle but is not the hypotenuse, so we will call this the adjacent side. This gives us the following.

From the above, we know that the quotients of the length of the side opposite 60 and the length of the side adjacent to 60 in any right triangle are equal. We can therefore define the functions that take an angle between 0 and 90 as an input and output of the values of these quotients to be the trigonometric ratios.

Definition: Trigonometric Ratios

The trigonometric ratios of an angle 𝜃 are the ratios of the side lengths in a right triangle. In particular, if we label the sides of any right triangle with an interior nonright angle 𝜃 as the hypotenuse, opposite, and adjacent, then sinoppositehypotenusecosadjacenthypotenusetanoppositeadjacent𝜃=,𝜃=,𝜃=.

Remembering exactly which sides correspond to which trigonometric ratio is quite difficult. So, to aid in this recollection, we use the acronym SOH CAH TOA.

Acronym: SOH CAH TOA

We can recall which sides correspond to each trigonometric function by using the acronym SOH CAH TOA. To do this, the first letter in each triplet corresponds to the trigonometric function, the second letter corresponds to the numerator of the quotient, and the third corresponds to the denominator of the quotient.

Once we have identified the relevant sides and hence the correct ratio for our right triangle, we can substitute any given values into the formula and solve to find either a side or an angle.

Let’s move on to our first example where we will need to label a triangle and use the trigonometric functions to determine an unknown side length.

Example 1: Finding the Unknown Length in a Right Triangle Using Trigonometry Where the Unknown Is on the Top of the Ratio

Find 𝑥 in the given figure. Give your answer to two decimal places.

Answer

We want to determine the length of an unknown side in a right triangle using a given side length and angle. We can do this by using the trigonometric functions. First, we need to determine which trigonometric function relates the unknown side and the side with length 11. We can do this by first labeling the sides based on their position relative to the angle 68. We note that the hypotenuse is opposite the right angle, 𝑥 is opposite the angle of 68 and so is the opposite side, and, finally, the remaining side adjacent to the angle is the adjacent side.

We notice that we have the length of the adjacent side, and we want to find the length of the opposite. This means we need to use the trigonometric ratio that links these two sides. We can recall which trigonometric function involves the ratio of these two sides by using the acronym SOH CAH TOA.

This allows us to identify that the tangent function is the ratio between the lengths of the side opposite the angle and the side adjacent to the angle in a right triangle. Therefore, we have tanAtan68=𝑂68=𝑥11.

We can solve for 𝑥 by multiplying through by 11: 𝑥=1168.tan

We can evaluate this expression by using a calculator where we must make sure the calculator is set to degrees mode: 𝑥=27.2259.

Rounding this value to two decimal places, we have 𝑥27.23.

In our previous example, we used a trigonometric function to determine an unknown side length in a right triangle by using a known side length and angle. We can follow this same process to determine unknown side lengths in any right triangle.

How To: Finding the Missing Side Length in a Right Triangle given an Angle and a Side Length

  1. If no diagram is given, sketch the given information onto a diagram.
  2. Label the sides of the right triangle based on their positions relative to the known angle.
  3. Use the acronym SOH CAH TOA to determine which trigonometric ratio includes the known side length and the side length required.
  4. Substitute the known values and rearrange to solve the equation for the missing side by using a calculator.

Let’s now see an example where we need to find two missing side lengths in a right triangle given an angle and a side length.

Example 2: Using Trigonometric Ratios to Find Two Missing Lengths of a Right Triangle

Find the values of 𝑥 and 𝑦 giving the answer to three decimal places.

Answer

We want to determine the length of two unknown sides in a right triangle using a given side length and angle. We recall that we can do this by using the trigonometric functions.

First, we need to determine which trigonometric function relates the unknown sides and the side with length 28 cm. We can do this by first labeling the sides based on their position relative to the angle 47.

We recall that the hypotenuse is the longest side and is opposite the right angle. This is the side of length 𝑦. The side opposite the angle is the side of length 28 cm, and the remaining side adjacent to the angle is the side of length 𝑥. This gives us the following labels.

We can now find equations involving the unknowns 𝑥 and 𝑦 by applying the trigonometric ratios to this triangle. We can recall which sides correspond to the trigonometric ratios by using the acronym SOH CAH TOA.

We want the trigonometric ratio linking the opposite and adjacent side to determine 𝑥 and the trigonometric ratio linking the opposite and hypotenuse to determine 𝑦. We see that these are the tangent and sine functions respectively.

Let’s start with determining the value of 𝑥: tanAtan47=𝑂47=28𝑥.

We rearrange for 𝑥 by multiplying through by 𝑥 and dividing through by tan47, which gives us 𝑥=2847.tan

We can evaluate this expression by using a calculator where we must make sure the calculator is set to degrees mode: 𝑥=26.1104.cm

Rounding this value to three decimal places, we get 𝑥26.110.cm

There are now several different methods we can use to find the value of 𝑦; we will go through two of these. First, we can use the sine function as follows: sinsin47=𝑂𝐻47=28𝑦.

We multiply through by 𝑦 and divide through by sin47 to get 𝑦=2847.sin

Then, we evaluate this expression: 𝑦=38.2851.cm

We can round this value to three decimal places to get 𝑦38.285.cm

The second method we can use to find 𝑦 is the Pythagorean theorem, which states that, in a right triangle, the square of the hypotenuse is equal to the sum of squares of the lengths of the two shorter sides. In this case, we have 𝑦=𝑥+28.

We found that 𝑥=2847tan and it is important to use the exact value to reduce rounding errors. Substituting this into the equation and simplifying, we get 𝑦=2847+28𝑦=2847+784𝑦=1465.7541.tantan

We then solve for 𝑦 by taking the square root of both sides of the equation, noting that 𝑦 is a length and so must be positive: 𝑦=1465.7541=38.2851, which agrees with our other calculation for 𝑦.

Hence, to three decimal places, 𝑥=26.110cm, 𝑦=38.285cm.

Let’s now see how to apply this process to solve a real-world problem.

Example 3: Using Right Triangle Trigonometry to Solve Word Problems

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 will be to sketch the information we are given. The angle with the horizontal is 60 and the perpendicular height of the kite is 44 m this gives us the following.

We want to determine the length of the string, which we have marked as 𝑥, since this is an unknown length in a right triangle and we have a known side length and a known angle. We can do this by using trigonometry.

First, we label the sides of the triangle based on their position relative to the angle of 60. We know that the side of length 𝑥 is opposite the right angle, so it is the hypotenuse. Next, we see that the side of length 44 m is opposite this angle and that the ground is the remaining side adjacent to the angle.

We know the length of the side opposite the angle and we want to find the length of the hypotenuse. We can recall which sides correspond to the trigonometric ratios by using the acronym SOH CAH TOA.

We want the trigonometric ratio linking the opposite sides and the hypotenuse. We can see this is the sine function.

We have sin60=𝑂𝐻=44𝑥.

Multiplying through by 𝑥, we get 𝑥60=44.sin

Dividing through be sin60 yields 𝑥=4460.sin

Now, we evaluate this expression by using a calculator: 𝑥=50.806.m

Finally, we round this value to one decimal place to get 𝑥50.8.m

In our next example, we will determine all of the unknown side lengths and the angle of a right triangle given only one side length and a nonright angle.

Example 4: Finding all the Unknowns in a Right Triangle

Given the following figure, find the lengths of 𝐴𝐶 and 𝐵𝐶 and the measure of 𝐵𝐴𝐶 in degrees. Give your answers to two decimal places.

Answer

To find the measure of 𝐵𝐴𝐶, we recall that the sum of the measures of the internal angles in a triangle is 180. Hence, 𝑚𝐵𝐴𝐶+𝑚𝐴𝐵𝐶+𝑚𝐴𝐶𝐵=180𝑚𝐵𝐴𝐶+90+22=180𝑚𝐵𝐴𝐶=1809022𝑚𝐵𝐴𝐶=68.

To determine the unknown side lengths, we will apply right triangle trigonometry. First, we need to label the sides of the triangle, and we note that 𝐴𝐶 is the longest side (since it is opposite the right angle), so this is the hypotenuse. Next, we will label the sides based on their position relative to the angle 22. We see that 𝐴𝐵 is opposite the angle and 𝐵𝐶 is adjacent to the angle, which gives us the following.

We now need to recall the trigonometric functions and we will do this by using the acronym SOH CAH TOA.

We know the length of the side opposite angle 𝐶, so we need both ratios using this value to determine the other side lengths.

First, we have sin22=𝑂𝐻=𝐴𝐵𝐴𝐶=4𝐴𝐶.

We multiply through by 𝐴𝐶 and divide through by sin22 to get 𝐴𝐶=422.sin

Evaluating this expression and rounding to two decimal places yields 𝐴𝐶10.68.cm

Second, we have tanA22=𝑂=𝐴𝐵𝐵𝐶=4𝐵𝐶.

We multiply through by 𝐵𝐶 and divide through by tan22 to get 𝐵𝐶=422.tan

Evaluating this expression and rounding to two decimal places yields 𝐵𝐶9.90.cm

Hence, to two decimal places, we have 𝐴𝐶=10.68cm, 𝐵𝐶=9.90cm, 𝑚𝐵𝐴𝐶=68.00.

In our next example, we will need to construct a diagram to apply right triangle trigonometry to find all the unknown angle and side lengths in a right triangle, given one side length and a nonright angle.

Example 5: Using Trigonometry to Solve Right Triangles with Angles in Degrees

𝐴𝐵𝐶 is a right triangle at 𝐵 where 𝑚𝐶=62 and 𝐴𝐶=17cm. Find the lengths of 𝐴𝐵 and 𝐵𝐶 giving the answer to two decimal places and the measure of 𝐴 giving the answer to the nearest degree.

Answer

We begin by sketching the information given onto a diagram. We know that the right angle is at 𝐵, 𝑚𝐶=62, and 𝐴𝐶=17cm. This gives us the following.

We can find 𝑚𝐴 by recalling that the sums of the measures of the internal angles in a triangle are 180. This gives 𝑚𝐴+𝑚𝐵+𝑚𝐶=180𝑚𝐴+90+62=180𝑚𝐴=1809062𝑚𝐴=28.

Next, we want to determine the length of two unknown sides in a right triangle using a given side length and angle. We recall that we can do this by using the trigonometric functions.

First, we need to determine which trigonometric function relates the unknown sides and the side with length 17 cm. We can do this by first labeling the sides based on their position relative to the angle 62.

We recall that the hypotenuse is the longest side and is opposite the right angle; this is 𝐴𝐶. The side opposite the angle is 𝐴𝐵, and the remaining side, adjacent to the angle, is 𝐵𝐶. This gives us the following labels.

We can now find equations involving 𝐵𝐶 and 𝐴𝐵 by applying the trigonometric ratios to this triangle. We can recall which sides correspond to the trigonometric ratios by using the acronym SOH CAH TOA.

We want the trigonometric ratio linking the opposite and hypotenuse to determine 𝐴𝐵 and the trigonometric ratio linking the adjacent and hypotenuse to determine 𝐵𝐶. We see that these are the sine and cosine functions respectively.

Let’s start with determining the value of 𝐴𝐵: sin62=𝑂𝐻=𝐴𝐵𝐴𝐶=𝐴𝐵17.

We rearrange for 𝐴𝐵 by multiplying through by 17: 𝐴𝐵=1762.sin

We can evaluate this expression by using a calculator where we must make sure the calculator is set to degrees mode: 𝐴𝐵=15.010.cm

Rounding this value to two decimal places, we get 𝑥15.01.cm

We can now determine the length of 𝐵𝐶 by using the cosine ratio: cosA62=𝐻=𝐵𝐶𝐴𝐶=𝐵𝐶17.

We then multiply through by 17 and evaluate as follows: 𝐵𝐶=1762=7.981.coscm

We can round this value to two decimal places to get 𝐵𝐶7.98.cm

Hence, to two decimal places, 𝐴𝐵=15.01cm, 𝐵𝐶=7.98cm, and, to the nearest degree, 𝑚𝐴=28.

In our next example, we will consider an isosceles triangle as two right triangles in order to determine the length of its base by using right triangle trigonometry.

Example 6: Using Trigonometry to Find the Length of the Base of an Isosceles Triangle

𝐴𝐵𝐶 is an isosceles triangle where 𝐴𝐵=𝐴𝐶=9cm, 𝐴𝐷𝐵𝐶 and 𝑚𝐶=34. Find the length of 𝐵𝐶 giving the answer to one decimal place.

Answer

We first note that 𝑚𝐴𝐷𝐶=90 since angles on a straight line add to make 180. We also note that the angles opposite the equal sides in an isosceles triangle are equal, so 𝑚𝐴𝐵𝐷=34. This means that 𝐴𝐷𝐶 and 𝐴𝐷𝐵 are two right triangles with the same internal angles, and 𝐴𝐶=𝐴𝐵. Hence, by the 𝐴𝑆𝐴 criterion, these triangles are congruent.

Since 𝐵𝐶=𝐵𝐷+𝐶𝐷, and these two line segments have equal lengths, we only need to determine the length of one of these line segments. In particular, we have 𝐵𝐶=𝐵𝐷+𝐶𝐷=𝐶𝐷+𝐶𝐷=2𝐶𝐷.

Since 𝐶𝐷 is a side of a right triangle with a known angle and side length, we can use trigonometry to determine its length. We start by labeling the sides of the triangle 𝐴𝐷𝐶. The hypotenuse is the longest side, which is opposite the right angle; in this case, it is 𝐴𝐶. The side opposite the angle 34 is 𝐴𝐷 and the remaining side adjacent to this angle is 𝐶𝐷. This gives us the following.

We can now use the acronym SOH CAH TOA to help us recall which trigonometric ratio is the quotient of the adjacent side and hypotenuse.

We see that this is the cosine function. Substituting the angle and side lengths into the formula gives us cosA34=𝐻=𝐶𝐷9.

Multiplying the equation through by 9 yields 𝐶𝐷=934.cos

We could evaluate; however, we want to find twice this value for the length 𝐵𝐶, so we instead multiply the equation through by 2 to get 𝐵𝐶=2𝐶𝐷=1834.cos

Evaluating gives us 𝐵𝐶=14.92.cm

Finally, rounding this value to one decimal place gives us 𝐵𝐶14.9.cm

In our final example, we will use two known angles in a triangle and a known side length to determine another side length.

Example 7: Dividing a Triangle into Right Triangles in Order to Calculate an Unknown Length

In the figure, 𝐴𝐶=3.5.

What is 𝐴𝐵? Give your answer to two decimal places.

Answer

Let’s first add the side length and the fact that 𝑚𝐴𝐷𝐶=90 to the diagram.

We want to determine 𝐴𝐵, but we have no side lengths in triangle 𝐴𝐷𝐵. Instead, we need to notice that we can use the known side length in the right triangle 𝐴𝐷𝐶 to determine 𝐴𝐷 by using trigonometry. We can then in turn use 𝐴𝐷 to find 𝐴𝐵.

Let’s start by finding 𝐴𝐷 using trigonometry. We need to label the sides of triangle 𝐴𝐷𝐶. First, 𝐴𝐶 is the longest side since it is opposite the right angle; this makes it the hypotenuse. Next, 𝐴𝐷 is opposite the angle of 41 and 𝐴𝐶 is the remaining side adjacent to this angle.

To determine 𝐴𝐷 we need to use the trigonometric function that is the quotient of the opposite side and the hypotenuse. We can recall this by using the acronym SOH CAH TOA.

Using this, we can recall that the sine function is the ratio between the length of the side opposite the angle and the hypotenuse. Substituting in the values from this right triangle, we get sin41=𝑂𝐻=𝐴𝐷𝐴𝐶=𝐴𝐷3.5.

We can then solve for 𝐴𝐷 by multiplying through by 3.5: 𝐴𝐷=3.541.sin

We could evaluate this expression; however, since we are going to use this to calculate another length, it is better to use the exact value.

To apply trigonometry to the right triangle 𝐴𝐷𝐵, we need to label its sides. 𝐴𝐵 is the hypotenuse since it is opposite the right angle, 𝐴𝐷 is opposite the angle of 63, and 𝐵𝐷 is the remaining side adjacent to the angle.

We want to find the length of the hypotenuse, and we know the length of the opposite side, so we will use the sine function: sinsin63=𝑂𝐻=𝐴𝐷𝐴𝐵=3.541𝐴𝐵.

We can solve for 𝐴𝐵 by multiplying through by 𝐴𝐵 and dividing through by sin63: 𝐴𝐵=3.54163.sinsin

Evaluating the expression gives us 𝐴𝐵=2.577.

Finally, rounding to two decimal places, we get 𝐴𝐵2.58.

Let’s finish by recapping some of the important points 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. 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 𝜃. The opposite is the side opposite angle 𝜃. Finally, the adjacent is the side next to the angle 𝜃 that is not the hypotenuse.
  • The trigonometric ratios of an angle 𝜃 are the ratios of the side lengths in a right triangle. In particular, if we label the sides of any right triangle with 𝜃 as an angle as the hypotenuse, opposite, and adjacent, then sinoppositehypotenusecosadjacenthypotenusetanoppositeadjacent𝜃=,𝜃=,𝜃=.
  • We can determine missing side lengths in a right triangle given a nonright angle and a side length by using the following steps.
    1. If no diagram is given, sketch the given information onto a diagram.
    2. Label the sides of the right triangle based on their position relative to the known angle.
    3. Use the acronym SOH CAH TOA to determine which trigonometric ratio includes the known side length and the side length we need to find.
    4. Substitute the known values and rearrange to solve the equation for the missing side by using a calculator.

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