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Lesson Explainer: Exact Values of Trigonometric Ratios Mathematics • 10th Grade

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.

π‘₯
0∘30∘45∘60∘90∘
sinπ‘₯012√22√321
cosπ‘₯1√32√22120
tanπ‘₯0√331√3Undefined

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 π‘₯ measured in degrees into one measured in radians by multiplying it by πœ‹180. This means that the 5 angles in our table can be written as 0Γ—πœ‹180=0,30Γ—πœ‹180=πœ‹6,45Γ—πœ‹180=πœ‹4,60Γ—πœ‹180=πœ‹3,90Γ—πœ‹180=πœ‹2.radradradradrad

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

π‘₯
0 radπœ‹6 radπœ‹4 radπœ‹3 radπœ‹2 rad
sinπ‘₯012√22√321
cosπ‘₯1√32√22120
tanπ‘₯0√331√3Undefined

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 tanπœ‹3=√3 since we know it is the same as tan60∘. 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 180∘ or πœ‹ rad. This means that tantan(π‘₯+π‘›πœ‹)=(π‘₯) for any integer 𝑛 and angle π‘₯ measured in radians.

We can use this identity and these known values of the trigonometric function to find more values. Consider tanο€Ό7πœ‹6; we know that 7πœ‹6=πœ‹+πœ‹6, so using the periodicity of the tangent function, we have tantantanο€Ό7πœ‹6=ο€»πœ‹+πœ‹6=ο€»πœ‹6.

We then note that tanο€»πœ‹6=√33. Hence, tanο€Ό7πœ‹6=√33.

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 cossin(90+π‘₯)=βˆ’(π‘₯)∘; in terms of radians, that is cossinο€»πœ‹2+π‘₯=βˆ’(π‘₯). We can also use this identity to find the exact value of the trigonometric function. If we set π‘₯=πœ‹4, then we have cossincosο€»πœ‹2+πœ‹4=βˆ’ο€»πœ‹43πœ‹4=βˆ’βˆš22.

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 cosο€»πœ‹3.

Answer

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 180πœ‹. We have πœ‹3Γ—180πœ‹=1803=60∘

We can then recall cos60∘. Hence, cosο€»πœ‹3=12.

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 cosο€»πœ‹3=12.

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 tan5πœ‹6.

Answer

We first note that the argument of the function 5πœ‹6 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: tantan(πœ‹βˆ’π‘₯)=βˆ’(π‘₯).

To use this identity, we need to first note that 5πœ‹6=πœ‹βˆ’πœ‹6. We can then substitute π‘₯=πœ‹6 into the identity to obtain tantantantanο€»πœ‹βˆ’πœ‹6=βˆ’ο€»πœ‹65πœ‹6=βˆ’ο€»πœ‹6.

We note that an angle of πœ‹6 radians is equivalent to an angle of πœ‹6Γ—180πœ‹=30∘ and that tan(30)=√33∘. Thus, βˆ’ο€»πœ‹6=βˆ’βˆš33tan.

Hence, tanο€Ό5πœ‹6=βˆ’βˆš33.

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 cos3πœ‹4.

Answer

We first note that the argument of the function 3πœ‹4 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: coscos(πœ‹βˆ’π‘₯)=βˆ’(π‘₯).

Next, we can observe that 3πœ‹4=πœ‹βˆ’πœ‹4. Therefore, we can then substitute π‘₯=πœ‹4 into the identity to obtain coscoscoscosο€»πœ‹βˆ’πœ‹4=βˆ’ο€»πœ‹43πœ‹4=βˆ’ο€»πœ‹4.

We now note that an angle of πœ‹4 radians is equivalent to an angle of πœ‹4Γ—180πœ‹=45∘ and that cos(45)=√22∘. Thus, βˆ’ο€»πœ‹4=βˆ’βˆš22cos.

Hence, cosο€Ό3πœ‹4=βˆ’βˆš22.

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 sinο€Ό13πœ‹6.

Answer

In order to determine the exact value of sinο€Ό13πœ‹6 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 13πœ‹6=12πœ‹+πœ‹6=2πœ‹+πœ‹6. Since the argument is more than a full rotation, we can rewrite it using the periodicity of the sine function, which is 360∘ or 2πœ‹ rad. Therefore, sinsin(π‘₯+2πœ‹π‘›)=π‘₯ for any angle π‘₯ in radians and any integer 𝑛.

Substituting 𝑛=1 and π‘₯=πœ‹6 into the identity yields sinsinsinsinο€»πœ‹6+2πœ‹(1)=πœ‹6ο€Ό13πœ‹6=πœ‹6.

We can now calculate that an angle of πœ‹6 rad is equivalent to an angle of πœ‹6Γ—180πœ‹=30∘, so sinsinπœ‹6=30=12.∘

Hence, sinο€Ό13πœ‹6=12.

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 sinο€Όβˆ’5πœ‹3.

Answer

In order to determine the exact value of sinο€Όβˆ’5πœ‹3 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 360∘ or 2πœ‹ rad. This means that sinsin(π‘₯+2πœ‹π‘›)=π‘₯ for any angle π‘₯ in radians and any integer 𝑛. If we set 𝑛=βˆ’1 in this identity, then we obtain sinsin(π‘₯βˆ’2πœ‹)=π‘₯.

We can then observe that πœ‹2βˆ’2πœ‹=βˆ’5πœ‹3, so if we substitute π‘₯=πœ‹3 into this identity, we have sinsinsinsinο€»πœ‹3βˆ’2πœ‹ο‡=ο€»πœ‹3ο‡ο€Όβˆ’5πœ‹3=ο€»πœ‹3.

We can evaluate sinπœ‹3 by noting that an angle of πœ‹3 rad is equivalent to an angle of πœ‹3Γ—180πœ‹=60∘, so sinsinο€»πœ‹3=60=√32.∘

Hence, sinο€Όβˆ’5πœ‹3=√32.

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.
    π‘₯
    0 radπœ‹6 radπœ‹4 radπœ‹3 radπœ‹2 rad
    sinπ‘₯012√22√321
    cosπ‘₯1√32√22120
    tanπ‘₯0√331√3Undefined
  • 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.

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