In this explainer, we will learn how to find and express the values of the three trigonometric ratios—sine, cosine, and tangent—for a given angle in a right triangle.

A useful skill when working with right triangles is having the ability to calculate the different trigonometric ratios for a specified angle. Recall the acronym “SOH CAH TOA.” This can be very useful in remembering the three different trigonometric ratios.

### Trigonometric Ratios

O stands for the “opposite,” A stands for the “adjacent,” H stands for the “hypotenuse,” and is the angle.

We can use the trigonometric ratios in combination with the Pythagorean theorem to calculate all the trigonometric ratios for a specified angle. We will demonstrate how to do this using a series of examples.

Before we start, let us recap how to correctly label a right triangle relative to a specified
angle. The **opposite** side is the side directly opposite the angle, the **hypotenuse**
is the longest side (which is always opposite the right angle), and the **adjacent** is the
side next to the angle (which is not the hypotenuse). We can demonstrate this using a
diagram.

It is helpful to recall the Pythagorean theorem. In words, it states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. For a triangle where the hypotenuse has length and the legs (the other two sides) have lengths and , the Pythagorean theorem can be represented algebraically as

Now, let us look at the first example.

### Example 1: Finding the Values of the Three Trigonometric Ratios of an Angle

Find the main trigonometric ratios of given that is a right triangle at , where and .

### Answer

Our first step in answering this question should be to draw a diagram of the triangle.

We then need to calculate the length of . We can do this using the Pythagorean theorem. We label the hypotenuse and the legs and . So, we have that , , and . Note here that the order of and does not matter as long as we are consistent throughout our calculation. Recall that

If we substitute the values of , , and , we get

We then subtract from each side and take square roots to find that

Calculating this, we find that so

We now need to add this length to our diagram and label the opposite, hypotenuse, and adjacent sides, relative to .

We can then use the acronym SOH CAH TOA to help us remember the trigonometric ratios and then we can use these to calculate the values of , , and .

By substituting the values of O, A, and H into the trigonometric ratios and simplifying the resulting fractions, we find that

Now, let us look at a question that contains an additional context and that adds an additional layer of difficulty.

### Example 2: Finding the Values of Trigonometric Ratios for an Angle given an Additional Context

is a diameter of a circle with radius 62.5 cm. Point is on the circumference of the circle where and . Find the exact values of and .

### Answer

The first thing to note here is that is a diameter and hence has a length that is twice the radius of the circle: . It is then helpful to draw the triangle using these lengths.

We then need to calculate the unknown length using the Pythagorean theorem. Let us call , , and the hypotenuse . Recall that

Substituting the lengths, we find that

If we then subtract from each side and take square roots, we get

Calculating this, we find that so

The question asks us to calculate and . We need to label the triangle in terms of the opposite, adjacent, and hypotenuse, first relative to and then relative to . When we label the triangle relative to , we get the following.

We can then use the acronym SOH CAH TOA to help us remember the trigonometric ratios.

Using these and the labeled triangle, we can calculate the value of :

If we then label the triangle relative to , we get the following.

Then, as before, we can calculate the value of :

In some questions, you will be given a specific trigonometric ratio and asked to calculate the value of others. To conclude, we will look at a couple of examples where this is the case.

### Example 3: Finding the Values of Trigonometric Ratios from a Known Ratio

The value of . Find the exact values of and .

### Answer

To solve this problem, it is helpful to start by drawing a right triangle containing an angle with labels for the opposite, adjacent, and hypotenuse, relative to .

We can then use the fact that to find that the opposite side has length 4 and the adjacent side has length 5. If we then call the length of the hypotenuse , we can find this using the Pythagorean theorem:

Taking square roots, we find that

Simplifying this, we find that

We will leave this in this form as the question asks for our answers in an exact form. We can now label the lengths of the triangle.

We can then recall the trigonometric ratios for sine and cosine, and use these, along with our triangle, to calculate our answers:

### Example 4: Finding the Values of Trigonometric Ratios from the Tangent Ratio

Find the main trigonometric ratios of given that is a right triangle at , where .

### Answer

To solve this problem, it is helpful to start by drawing a right triangle. We are told in the question that . We can rearrange this to give us and we can compare this to the general tangent ratio

We find that the side opposite to angle has length 21 and the side adjacent to angle has length 20.

We can now find the length of the hypotenuse using the Pythagorean theorem. If we call the length of the hypotenuse , we have

Taking square roots, we find that

Simplifying this, we find that

We can now label the hypotenuse in our diagram and label the sides relative to .

We can then recall the trigonometric ratios for sine and cosine, and use these, along with our triangle and tangent ratio, to calculate our answers:

The methods outlined in these latter two examples become very useful when working with questions in mechanics that involve slopes or ramps. The angle of the slope is often given as a tangent ratio of its length and height, and calculations will often require the use of the sine and cosine ratios.

Let us look at one more example where we need to use the trigonometric ratios to help solve the problem.

### Example 5: Using Trigonometric Ratios to Solve Problems

Find given that is a right triangle at , where .

### Answer

From the question, we are told that the triangle is a right triangle at and we are asked to calculate the measure of angle . We can use this information to draw a diagram, labeling the triangle with respect to angle :

We are also told information about lengths and , specifically, that

Rearranging, we can see that

Looking at the diagram, we can see that this is a ratio between the hypotenuse and the adjacent sides of the triangle. Recall that the cosine ratio is which, for this triangle, is

This is the reciprocal of what we were given in the question. That is,

Therefore, which means

### Key Points

To find the values of the three trigonometric ratios of an angle, remember the following steps:

- 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 relationship to a given angle often denoted . The adjacent is the side next to the angle 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. - Recall the acronym “SOH CAH TOA,” where O stands for the opposite, A stands for the adjacent, H stands for the hypotenuse, and is the angle. The trigonometric ratios are
- We can use the trigonometric ratios in combination with the Pythagorean theorem, to calculate all the trigonometric ratios for a specified angle.