# Lesson Video: Right Triangle Trigonometry: Solving for a Side Mathematics

In this video, 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.

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### Video Transcript

In this video, 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. Suppose we have a right triangle like the one shown with one of the nonright angles labeled as 𝜃. The hypotenuse of a right triangle is its longest side, which is always the side directly opposite the right angle. In relation to the angle we’ve labeled as 𝜃, the side directly opposite this angle is known as the opposite. And finally, the other side next to the angle 𝜃, which isn’t the hypotenuse, is called the adjacent. This is also the side that is between the right angle and 𝜃.

We’ll often see the names for these three sides abbreviated to opp, adj, and hyp or simply O, A, and H. The three trigonometric ratios sine, cosine, and tangent, which we abbreviate to sin, cos, and tan, describe the ratios between different pairs of side lengths in a right triangle. For a fixed value of 𝜃, the ratio between each pair of side length is always the same, no matter how big the triangle is. We can use the acronym SOHCAHTOA to help us remember the definitions of the three trigonometric ratios. The first letter in each part refers to either sine, cosine, or tangent. And then the next two letters refer to the sides involved in the ratio, the numerator first and then the denominator.

So SOH tells us that the sine ratio, sin of a given angle 𝜃, is equal to the length of the opposite side divided by the length of the hypotenuse. cos of 𝜃 is equal to the length of the adjacent side divided by the length of the hypotenuse. And tan of 𝜃 is equal to the length of the opposite side divided by the length of the adjacent. Let’s now look at how we can use these trigonometric ratios to calculate an unknown length in a right triangle.

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

In this right triangle, we have been given the length of one side and the measure of one of the other angles. We want to calculate the length of another side. We’ll begin by labeling the size of the triangle in relation to the angle of 68 degrees. The side directly opposite the right angle is always the hypotenuse, which we’ll abbreviate to H. The side directly opposite the angle of 68 degrees is the opposite side, which we’ll abbreviate to O. And finally, the side between the right angle and the angle of 68 degrees is the adjacent, which we’ll abbreviate to A. We can use the acronym SOHCAHTOA to help us decide which of the trigonometric ratios we need to use in this question.

The side we wish to calculate is the opposite and the side whose length we know is the adjacent, which tells us we’re going to be using the tangent ratio. Let’s recall its definition. In a right triangle, the tan of an angle 𝜃 is equal to the length of the opposite side divided by the length of the adjacent side. Now we substitute the value of 𝜃 and the values or expressions for the opposite and adjacent in this triangle. 𝜃 is 68 degrees. The opposite we don’t know, but we have the expression 𝑥. And the adjacent is 11 units. So we have the equation tan of 68 degrees is equal to 𝑥 over 11.

To solve this equation for 𝑥, we need to multiply both sides by 11. This gives 11 multiplied by tan of 68 degrees is equal to 𝑥. Or we could just write this as 11 tan 68 degrees. We don’t need the multiplication sign. We can now evaluate this on our calculators, making sure they are in degree mode. And it gives 27.225. The question specifies that we should give our answer to two decimal places. So rounding our answer, we have 27.23. No units were given in the question, so our answer is simply 27.23 length units.

In this example, the unknown we wanted to calculate was in the numerator or top of the fraction. And so rearranging our equation to solve for 𝑥 was quite straightforward. Let’s now look at another example where the unknown is in the denominator of the fraction.

Find 𝑥 to two decimal places.

In this right triangle, we know the measure of one of the other angles and the length of one side. We want to calculate the length of another side of this triangle. We can do this using trigonometry. We’ll begin by labeling the three sides of this triangle in relation to the angle of 20 degrees. The side directly opposite the right angle is the hypotenuse of the triangle. The side opposite the angle of 20 degrees is the opposite. And the side between the right angle and the angle of 20 degrees is the adjacent.

Next, we recall the acronym SOHCAHTOA to help us decide whether we need the sine, cosine, or tangent ratio in this question. The side we know is the opposite. And the side we want to calculate is the hypotenuse. So we’re going to be using the sine ratio. For a given angle 𝜃 in a right triangle, the sine ratio, sin of 𝜃, is equal to the length of the opposite side divided by the length of the hypotenuse. We can substitute the values for this triangle into this definition. 𝜃 is equal to 20 degrees, the opposite is 12 units, and the hypotenuse is this unknown 𝑥. So we have the equation sin of 20 degrees is equal to 12 over 𝑥.

Now, we must be careful here. A really common mistake is to think that the unknown, in this case 𝑥, must always be in the numerator of the fraction, and so to write down instead sin of 20 degrees is equal to 𝑥 over 12. But of course, if we did that, we would be dividing the length of the hypotenuse by the length of the opposite, not the length of the opposite by the length of the hypotenuse. This is a really common mistake, though. So we just need to take our time when substituting the values or expressions for each side of the triangle into the definition of our trigonometric ratios. We now need to solve this equation where 𝑥 appears in the denominator of the fraction, and this will require two steps.

First, we multiply both sides of the equation by our unknown 𝑥. On the left-hand side, we now have 𝑥 sin 20 degrees and on the right-hand side, 12 over 𝑥 multiplied by 𝑥 simplifies to 12. Next, we need to divide both sides of the equation by sin of 20 degrees. Remember, this is just a number, so it’s absolutely fine to do this. This gives 𝑥 is equal to 12 over sin of 20 degrees. Finally, we evaluate on our calculators, giving 35.085. Remember, we must make sure that our calculators are in degree mode in order to give the correct answer. The question specifies that we should give our answer to two decimal places. So we round to 35.09. So by applying the sine ratio in this right triangle, we found that the value of 𝑥 to two decimal places is 35.09.

We’ve now seen examples of how to calculate an unknown side length both when it appears in the numerator and the denominator of a fraction. Let’s summarize the key steps we need to perform. First, we label the three sides of the triangle using the letters O, A, and H to represent the opposite, adjacent, and hypotenuse. Next, we identify the side whose length we know and the side we wish to calculate. Then we use the acronym SOHCAHTOA to help us decide which trigonometric ratio we need to use.

We then write down the definition of that trigonometric ratio and substitute the values for the particular triangle we’re working with. Finally, we solve the equation to find the missing length and evaluate using our calculator. Remember, this will sometimes require a more complex rearrangement if the side we need to calculate is in the denominator of the fraction.

In each of the problems we’ve considered so far, we’ve been given a diagram of the triangle we need to use. That may not always be the case. So let’s look at a question in which we first need to sketch the triangle from a worded description.

𝐴𝐵𝐶 is a right-angled triangle at 𝐵 where the measure of angle 𝐶 is 62 degrees and 𝐴𝐶 is 17 centimeters. Find the lengths of 𝐴𝐵 and 𝐵𝐶 giving the answer to two decimal places and the measure of angle 𝐴 giving the answer to the nearest degree.

Let’s begin by drawing a sketch of this triangle. We’re told that it is right-angled at 𝐵. So 𝐵 is the vertex by the right angle and the other two vertices are 𝐴 and 𝐶. The other information we’re given is that the measure of angle 𝐶 is 62 degrees and 𝐴𝐶 is 17 centimeters. We’re asked to find the length of 𝐴𝐵 and 𝐵𝐶. Those are the other two sides of the triangle. So we’ll call them 𝑥 centimeters and 𝑦 centimeters. And we’ll also ask to find the measure of angle 𝐴.

Now, actually, we can work out the measure of angle 𝐴 straightaway because we have a triangle in which we know the other two angles. The angle sum in any triangle is 180 degrees, so we can work out the measure of the third angle by subtracting the other two from 180 degrees. That gives 28 degrees. Now, let’s think about how we’re going to find the lengths of the other two sides of this triangle. We’ll begin by labeling all three sides in relation to the angle of 62 degrees. 𝐴𝐶 is the hypotenuse, 𝐴𝐵 which we’re calling 𝑥 centimeters is the opposite, and 𝐵𝐶 is the adjacent.

We’ll then recall the acronym SOHCAHTOA to help us decide which trigonometric ratio we need to calculate the length of each side. Starting with 𝐴𝐵, first of all, the side we want to calculate is the opposite, and the side we know is the hypotenuse. So we’re going to use the sine ratio. This tells us that sin of an angle 𝜃 is equal to the opposite divided by the hypotenuse. Substituting the values for this triangle, we have sin of 62 degrees is equal to 𝑥 over 17. We solve for 𝑥 by multiplying both sides of the equation by 17 giving 𝑥 equals 17 sin 62 degrees. Evaluating gives 15.0101 which we round to 15.01.

To calculate the second side, 𝐵𝐶, we have a choice. As we now know the length of two sides in this right triangle, we could calculate the length of the third side by applying the Pythagorean theorem. But as we’re focusing on trigonometry here, let’s instead calculate 𝐵𝐶 using the trigonometric ratios. This time, the side we want to calculate is the adjacent and the side we were originally given is the hypotenuse. So we’re going to use the cosine ratio. Alternatively, we could use the side we’ve just calculated, which would give the pair O and A. So we’d be using the tan ratio. But it makes sense to use the value we were originally given in case you made any mistakes when calculating the length of the opposite.

Substituting 62 degrees for 𝜃, 𝑦 for the adjacent, and 17 for the hypotenuse gives cos of 62 degrees equals 𝑦 over 17. We can then multiply both sides of the equation by 17 to give 𝑦 equals 17 cos 62 degrees and evaluate on our calculators, making sure they’re in degree mode. We then round to two decimal places, giving 7.98. So we’ve completed the problem. The length of 𝐴𝐵 is 15.01 centimeters. The length of 𝐵𝐶 is 7.98 centimeters, each to two decimal places. And the measure of angle 𝐴 is 28 degrees.

Right-triangle trigonometry is really useful because it can also be applied in practical contexts. Often, the problems we face will take the form of a story or a description of a real-life situation and can be solved by applying the techniques we’re practicing here. If we aren’t given a diagram, then a key first step will be for us to draw a sketch based on the information we’re given. Let’s look at one final example of this type.

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

Let’s begin by drawing a sketch of this problem. We have a kite which is attached to a string. This string is inclined at an angle of 60 degrees to the horizontal and the perpendicular height of the kite. So that means the height of the kite that makes a right angle with the horizontal is 44 meters. We can now see that we have a right triangle formed by the horizontal, the vertical, and the string of the kite. We want to calculate the length of the string, so let’s label that as 𝑦 meters. We’re working with a right triangle, so we can approach this problem using trigonometry.

We’ll begin by labeling the three sides of the triangle in relation to the angle of 60 degrees. Next, we’ll recall the acronym SOHCAHTOA to help us decide which trigonometric ratio we need here. The side whose length we know is the opposite, and the side we want to calculate is the hypotenuse. So we’re going to be using the sine ratio. For an angle 𝜃 in a right triangle, this is defined as the length of the opposite divided by the length of the hypotenuse. We can then substitute the values for 𝜃, the opposite and the hypotenuse, into this equation giving sin of 60 degrees equals 44 over 𝑦.

We need to be careful because the unknown appears in the denominator of this fraction. Next, we solve this equation. As 𝑦 appears in the denominator, the first step is to multiply both sides of the equation by 𝑦, which gives 𝑦 sin 60 degrees is equal to 44. Next, we divide both sides of the equation by sin of 60 degrees, giving 𝑦 equals 44 over sin of 60 degrees. And then we evaluate on our calculators, which must be in degree mode, giving 50.806. The question asks for our answer accurate to one decimal place. So we round this value and include the units which are meters. The length of the string to one decimal place is 50.8 meters.

Let’s now summarize the key points from this video. When working with right triangles, we use the terms opposite, adjacent, and hypotenuse to refer to the three sides of the triangle. The hypotenuse is directly opposite the right angle and it is always the longest side of a triangle. The opposite and adjacent are labeled in relation to a given angle, often denoted 𝜃. The opposite is the side directly opposite this angle, whereas the adjacent is the side between this angle and the right angle. We can use the acronym SOHCAHTOA to help us determine which trigonometric ratio we need to use to calculate a missing side. Sin of 𝜃 is equal to the opposite over the hypotenuse. cos of 𝜃 is equal to the adjacent over the hypotenuse. And tan of 𝜃 is equal to the opposite over the adjacent.

When using trigonometry to find an unknown side length in a right triangle, we work through the following steps. First, we label the sides of the triangle in relation to the known angle 𝜃. Secondly, we use SOHCAHTOA to choose the correct trigonometric ratio. We then substitute the known angle and the known side length. And finally, we solve the equation to calculate the unknown. Remember, we need to take extra care when rearranging if the unknown is in the denominator of the fraction. We also saw that we can apply these techniques to worded problems that describe a real-life situation.