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.
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 describe, we will first consider a few geometric properties of right triangles that will allow us to derive the formulas for each ratio.
First, we recall that the sum of the measures of the interior angles in a triangle is . This means, if we have a right triangle with a nonright angle of , the remaining angle always has measure equal to .
Second, we recall that scaling a shape by a scale factor of does not affect the measure of its angles and enlarges all of the side lengths by a multiplicative factor of .
These two facts allow us to notice an interesting property of right triangles: the ratios of the lengths of the sides of a right triangle are only dependent on the angle and the choice of the two sides. To help us understand this, let’s consider the following triangle.
We see that is a right triangle and , so we can deduce that . If we were to scale this triangle by a factor of , the angles would stay the same and all the lengths would be multiplied by this factor of , giving us the following triangle.
We can then note that the ratios of the corresponding sides are unaffected by this dilation. For example,
Therefore, all right triangles with an angle of will have equal ratios of their corresponding side lengths. Of course, there was nothing unique about the choice of ; this result would be true for any angle we had chosen.
Before we state this result and discuss its uses, we need to determine 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 that easy. So, instead, we label each of the sides of the triangle based on their positions relative to the angle. We also recall that the angle between two sides that meet at a vertex is called the included 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 right triangle , we see that 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 . We see that is opposite the angle, so we will call this the opposite side, and 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 to and the length of the side adjacent to in any right triangle are equal. We can extend this idea to include any angle in the interval to and relate any pair of sides this way. Let’s define our functions as taking an angle between and as their input and outputting the value of the quotient between a pair of the side lengths.
We give these functions a name depending on which quotient they describe. For instance, the sine ratio links the opposite with the hypotenuse, while the cosine ratio links the adjacent with the hypotenuse. Let’s describe these formally.
Definition: Trigonometric Ratios
The trigonometric ratios, sine, cosine, and tangent, of an angle are the ratios of the side lengths in a right triangle. In particular, if we label the sides of any right triangle with as an angle as the hypotenuse, opposite, and adjacent, then
Remembering exactly which sides correspond to which trigonometric ratio can be 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.
Let’s move on to our first example where we will determine the values of all three trigonometric ratios of an angle, given two side lengths of the right triangle.
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
We begin by sketching the right triangle with the information given. The right angle is at , , and .
We recall that the trigonometric ratios of an angle are the ratios of the side lengths in a right triangle, so we need to find the side lengths and label the sides of this triangle based on their positions relative to .
Let’s start by labeling the sides of the triangle. First, we note that is the longest side of the triangle since it is opposite the right angle; hence, it is the hypotenuse. Next, we see that is opposite , so this is the opposite side. Finally, the remaining side is the adjacent side, so this gives us the following.
Before we determine the trigonometric ratios for , we need to find the length of . We can do this by applying the Pythagorean theorem. The Pythagorean theorem tells us that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the lengths of the two shorter sides. In this case, we get
We can solve for by first subtracting 324 from both sides of the equation to get
Next, we take the square root of both sides of the equation where we note that is a length and so must be nonnegative. This yields
We can add this length to the diagram.
We are now ready to determine the trigonometric ratios of . We recall that we can use the acronym SOH CAH TOA to help us remember which sides we take as ratios.
This helps us recall that the sine of an angle is the ratio between the length of the side opposite angle and the hypotenuse of the right triangle. Similarly, the cosine of an angle is the ratio between the length of the side adjacent to angle and the hypotenuse of the right triangle. Finally, the tangent of an angle is the ratio between the length of the side opposite angle and the length of the side adjacent to angle . We substitute each of the lengths into these formulas to evaluate the ratios:
Hence, , , and .
In our next example, we will determine the values of two trigonometric ratios of an angle given two side lengths of the right triangle in order to evaluate a trigonometric expression.
Example 2: Finding the Value of Trigonometric Ratios Given Two Sides of the Right Triangle
Find given that is a right triangle at , where and .
Answer
We want to determine the values of and . To do this, we recall that the trigonometric ratios of an angle are the ratios of side lengths in a right triangle. In particular, we can use the acronym SOH CAH TOA to help us remember which sides we take as ratios.
Therefore, to determine the values of and , we need to label the sides of the right triangle based on their positions relative to angle . We do this by first noting that the longest side of the right triangle is the one opposite the right angle. In this triangle, this is side , which is therefore the hypotenuse. Next, we label the side opposite angle and the side adjacent to angle , which gives us the following triangle.
We know that . However, we cannot determine the value of since we do not know the length of the adjacent side to angle . We can find the missing side length by recalling that the Pythagorean theorem tells us that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the lengths of the two shorter sides. In this case, we get
We then subtract 64 from both sides of the equation to get
Then, we take the square root of both sides of the equation, where we note that is a length and hence is nonnegative
We can then substitute this value into the formula for the cosine ratio to get
Finally, we can use these values to evaluate the given expression
Example 3: Finding the Values of Trigonometric Ratios for Two Different Angles
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
First, we are told that the radius of the circle is 62.5 cm and that is a diameter, so has length twice the radius: . Let’s now isolate triangle and add the length of to the diagram.
We want to determine the values of and . To do this, we recall the acronym SOH CAH TOA.
Therefore, to determine the values of and , we need to label the sides of the right triangle based on their positions relative to each angle. Let’s start with angle ; the side opposite the right angle, which is the longest side of the right triangle called the hypotenuse; the side opposite angle , which is called the opposite side; and the side adjacent to angle that is not the hypotenuse, which is called the adjacent side. This gives us the following triangle.
Now, since and we know the lengths of the side adjacent to angle and the hypotenuse, we can determine the value of :
We can follow the same process to find . First, we label the sides of the triangle based on their positions relative to angle . We note that the hypotenuse is the side opposite the right angle, is the side opposite angle , and is the side adjacent to angle . We have the following triangle.
Since , we have
This is enough to answer the question; however, it is worth noting that we can use the Pythagorean theorem to determine the unknown side length in the right triangle. We recall that the Pythagorean theorem tells us that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the lengths of the two shorter sides. In this case, we get
We then subtract 5 625 from both sides of the equation to get
Then we take the square root of both sides of the equation, where we note that is a length and so must be nonnegative:
We could then use this to determine any of the other trigonometric ratios.
Hence, and .
Example 4: Finding the Main Trigonometric Ratios in a Triangle Split into Two Right Triangles
Find the value of , given that , , , and .
Answer
We are asked to find the value of an expression involving trigonometric ratios in the given diagram. To do this, we recall the acronym SOH CAH TOA.
In particular, since we need to find trigonometric ratios for two different angles, we will need to label the sides of each triangle relative to both angles.
Let’s start with in right triangle . First, the hypotenuse is the longest side in the right triangle, which is the side opposite the right angle; in this case, that is . Second, the side opposite is the opposite side, that is . Third, the remaining side is the adjacent side, which is . We can update the given diagram with these labels.
We know that ; however, we do not know the length of . To find this length, we will use the Pythagorean theorem, which states that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the lengths of the two shorter sides. Applying this to triangle gives us
We can solve for by first subtracting 784 from both sides of the equation, giving us
Then, we take the square root of both sides of the equation where we note that is a length and is hence nonnegative:
Hence,
We now want to apply the same process to triangle to determine the value of . We can add to the diagram and label the sides of this triangle by noting that is opposite the right angle, is opposite , and is adjacent to .
We can now evaluate as follows:
Finally, we can substitute these values into the given expression:
In our final two examples, we will use a given trigonometric ratio to determine the values of other trigonometric ratios in the right triangle.
Example 5: Finding the Values of Cosine and Sine given the Value of Tangent
Find the main trigonometric ratios of given that is a right triangle at , where .
Answer
We start by rearranging the given equation to make the tangent ratio the subject, and we divide through by 20 to get
We then recall that the trigonometric ratios of an angle are the ratios of side lengths in a right triangle. We can use the acronym SOH CAH TOA to help us remember which side-length ratios correspond to each of the trigonometric functions.
In particular, we can note that the tangent function is the ratio of the length of the side opposite the angle and the length of the side adjacent to the angle in a right triangle. We know that , so we might want to conclude that the and the . However, this is a ratio, so any multiple of these values will also hold true. For example, we could have the and the , then
Hence, we cannot determine the exact lengths of these sides. Instead, we will say that the length of the opposite side to is 21 multiplied by a certain positive factor and that the length of the adjacent side to is 20 multiplied by the same positive factor. We can then sketch the following right triangle.
We are told has a right angle at , and we know that is opposite and is adjacent to . Finally, we add in the lengths: and .
We cannot determine the sine and cosine ratios of this right triangle at without finding an expression for the hypotenuse. We can find this by using the Pythagorean theorem, which states that in a right triangle the square of the hypotenuse is equal to the sum of the squares of the lengths of the two shorter sides. In this right triangle, this gives us
We can then solve for by taking the square root of both sides of the equation where we note that is a length; therefore, it must be positive and . This yields
We can then add this to our diagram along with labeling this side as the hypotenuse. We then label the sides of the triangle based on their positions relative to angle . This gives us the following triangle.
We are now ready to determine the values of , , and . Firstly,
Secondly,
Thirdly,
Hence, , , and .
Example 6: Finding the Main Trigonometric Ratios given the Ratio of Two Sides of the Right Triangle
Find the main trigonometric ratios of , given that is a right triangle at , where the ratio between and is .
Answer
We can start by sketching triangle , where we know the right angle is at vertex . Since the question wants us to determine the trigonometric ratios of , we can also label the sides based on their positions relative to .
We know that is opposite angle , is opposite the right angle and is therefore the hypotenuse, and is adjacent angle .
Therefore, since we are told the ratio between and is , we also know that the ratio between the lengths of the adjacent side to and hypotenuse is . In other words,
We can use the acronym SOH CAH TOA to help us remember which side-length ratios correspond to each of the trigonometric functions.
We know that , so . We cannot use this to directly find the lengths of the sides of this triangle since we only know the ratio between the lengths. We can, however, say that and for some positive value of .
We can use these lengths and the Pythagorean theorem to find an expression for . We recall that the Pythagorean theorem states that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the lengths of the two shorter sides. Hence,
We can solve for by taking the square root of both sides of the equation where we note that is a length, and so it must be positive, and that . Therefore, we get
We can add these lengths to our diagram.
We can now determine the remaining trigonometric ratios for .
First,
Second,
Hence, , , and .
Let’s finish by recapping some of the important points of this explainer.
Key Points
- To find the values of the three trigonometric ratios of an angle, we remember the following steps:
- Label the sides of the triangle based on their positions relative to the angle whose trigonometric ratios we want to calculate. The hypotenuse is always opposite the right angle and is the longest side, the opposite side is the side opposite the angle, and the adjacent side is the remaining side adjacent to the angle.
- Recall the acronym SOH CAH TOA where O stands for the opposite, A stands for the adjacent, and H stands for the hypotenuse. If the angle is , this helps us recall that the trigonometric ratios are
- If we only know two side lengths in a right triangle, then we can use the Pythagorean theorem to determine the final side length. This allows us to determine all of the trigonometric ratios of a right triangle from one angle and two side lengths.