 Lesson Explainer: Exact Values of Trigonometric Ratios | Nagwa Lesson Explainer: Exact Values of Trigonometric Ratios | Nagwa

# Lesson Explainer: Exact Values of Trigonometric Ratios Mathematics

In this explainer, we will learn how to find the exact value of a trigonometric function for radian angles.

We have found the exact values of the trigonometric ratios for angles in degrees. The important ones to recall are listed in the following table.

01
10
01Undefined

We found these values either by using the definition of the trigonometric ratios using a unit circle centered at the origin or by constructing right triangles with known angles and side lengths.

We could follow these processes again to find the exact values of the trigonometric ratios in radians. However, we do not need to do this; we can instead just convert the angles we have already found into radians. This would then give us a table of exact values for angles measured in radians.

We can convert an angle measured in degrees into one measured in radians by multiplying it by . This means that the 5 angles in our table can be written as

This gives us the following table for the exact values of trigonometric functions for angles measured in radians.

### Property: Exact Values of Trigonometric Ratios for Angles Measured in Radians

01
10
01Undefined

This allows us to easily evaluate any trigonometric functions at these values by recalling the table or even their values in degrees. For example, we can just state that since we know it is the same as . We sometimes refer to these as special angles since we can easily evaluate the trigonometric functions at these angles.

We have seen how to use these known values with various identities to determine the exact values of trigonometric functions at other values. For example, we know that the tangent function is periodic with a period of or rad. This means that for any integer and angle measured in radians.

We can use this identity and these known values of the trigonometric function to find more values. Consider ; we know that , so using the periodicity of the tangent function, we have

We then note that . Hence,

We can combine the results in the table with other trigonometric identities as well. For example, we can recall that the cofunction identities tell us that ; in terms of radians, that is . We can also use this identity to find the exact value of the trigonometric function. If we set , then we have

In our first example, we will evaluate a trigonometric function at a special angle measured in radians.

### Example 1: Finding the Exact Value of the Cosine Ratio Using Radians

Find the value of .

One way we can answer this question is by first converting the angle in radians into one in degrees. We do this by multiplying it by . We have

We can then recall . Hence,

Of course, it is useful to commit the conversions of useful angles and the exact values of the trigonometric ratios at these special angles to memory.

In either case, we have

In our next example, we will use an identity to evaluate a trigonometric function without a calculator by using an identity.

### Example 2: Finding the Exact Value of the Tangent Ratio Using an Identity

Find the value of .

We first note that the argument of the function is not a standard angle, so we cannot directly evaluate this expression. Instead, we are going to need to use an identity to rewrite the expression in terms of angles we do know how to evaluate.

We can do this by recalling the following identity for the tangent function:

To use this identity, we need to first note that . We can then substitute into the identity to obtain

We note that an angle of radians is equivalent to an angle of and that . Thus, .

Hence,

In our next example, we will determine the exact value of a trigonometric function at an angle measured in radians by applying a trigonometric identity.

### Example 3: Using Cofunction Identities to Rewrite and Find Trigonometric Ratios

Without using a calculator, determine the exact value of .

We first note that the argument of the function is not a standard angle, so we cannot directly evaluate this expression. Instead, we are going to need to use an identity to rewrite the expression in terms of angles we do know how to evaluate.

We can do this by recalling the following identity for the cosine function:

Next, we can observe that . Therefore, we can then substitute into the identity to obtain

We now note that an angle of radians is equivalent to an angle of and that . Thus, .

Hence,

In our next example, we will use a periodic identity to find an exact value of a trigonometric ratio at an angle measured in radians.

### Example 4: Using Identities to Rewrite and Find Trigonometric Ratios

Without using a calculator, determine the exact value of .

In order to determine the exact value of without using a calculator, we need to rewrite the expression in terms of angles we do know how to evaluate. To do this, we can first note that . Since the argument is more than a full rotation, we can rewrite it using the periodicity of the sine function, which is or rad. Therefore, for any angle in radians and any integer .

Substituting and into the identity yields

We can now calculate that an angle of rad is equivalent to an angle of , so

Hence,

In our final example, we will determine the exact value of a trigonometric function whose argument is a negative angle measured in radians.

### Example 5: Using Trigonometric Identities to Rewrite and Find Trigonometric Ratios

Without using a calculator, determine the exact value of .

In order to determine the exact value of without using a calculator, we need to rewrite the expression in terms of angles we do know how to evaluate. There are many different ways we can do this; however, we will only go through one of these methods.

First, we can recall that the sine function has a periodicity of or rad. This means that for any angle in radians and any integer . If we set in this identity, then we obtain

We can then observe that , so if we substitute into this identity, we have

We can evaluate by noting that an angle of rad is equivalent to an angle of , so

Hence,

Let’s finish by recapping some of the important points from this explainer.

### Key Points

• We can use the known exact values of the trigonometric functions at angles measured in degrees and the conversion formula to find the exact values of trigonometric functions at angles measured in radians. We can write these in the following table.
01
10
01Undefined
• Every identity for the trigonometric functions in degrees also holds for angles measured in radians after a conversion. We can use these identities in combination with the exact values above to determine the exact values of the trigonometric functions at other arguments.