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Mathematics LibreTexts

6.3 Inverse Trig Functions

In previous sections we have evaluated the trigonometric functions at various angles, but at times we need to know what angle would yield a specific sine, cosine, or tangent value.  For this, we need inverse functions.  Recall that for a one-to-one function, if File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image411.gif, then an inverse function would satisfy File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image413.gif.

You probably are already recognizing an issue – that the sine, cosine, and tangent functions are not one-to-one functions.  To define an inverse of these functions, we will need to restrict the domain of these functions to yield a new function that is one-to-one.  We choose a domain for each function that includes the angle zero.

 

Sine, limited to File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image415.gif       Cosine, limited to File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image417.gif          Tangent, limited to File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image419.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image421.jpg         File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image423.jpg         File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image425.jpg

On these restricted domains, we can define the inverse sine, inverse cosine, and inverse tangent functions.

 

Inverse Sine, Cosine, and Tangent Functions

For angles in the interval File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image427.gif, if File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image429.gif, then File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image431.gif

For angles in the interval File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image417.gif, if File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image434.gif, then File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image436.gif

For angles in the interval File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image438.gif, if File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image440.gif, then File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image442.gif

 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image444.gif has domain [-1, 1] and range File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image427.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image446.gif has domain [-1, 1] and range File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image417.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image448.gif has domain of all real numbers and range File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image438.gif

 

The File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image444.gif is sometimes called the arcsine function, and notated File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image451.gif.

The File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image446.gif is sometimes called the arccosine function, and notated File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image453.gif.

The File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image448.gif is sometimes called the arctangent function, and notated File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image455.gif.

 

The graphs of the inverse functions are shown here:

 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image444.gif                                  File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image446.gif                                 File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image448.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image457.jpg          File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image459.jpg          File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image461.jpg

 

Notice that the output of each of these inverse functions is an angle

Example 1
 

 

 

 

 

Try It Now

1. Evaluate

a) File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image497.gif              b) File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image499.gif                c) File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image501.gif                d) File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image503.gif

 

Example 2

Evaluate

a)  File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image463.gif             b) File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image465.gif          c) File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image467.gif          d) File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image469.gif

 

a) Evaluating File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image463.gif is the same as asking what angle would have a sine value of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image026.gif.  In other words, what angle θ would satisfy File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image473.gif?  There are multiple angles that would satisfy this relationship, such as File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image475.gif and File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image477.gif , but we know we need the angle in the  interval File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image427.gif, so the answer will be File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image479.gif.  Remember that the inverse is a function so for each input, we will get exactly one output.

 

b) Evaluating File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image465.gif, we know that File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image481.gif and File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image483.gif both have a sine value of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image485.gif, but neither is in the interval File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image427.gif.  For that, we need the negative angle coterminal with File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image483.gifFile:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image488.gif.

c) Evaluating File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image467.gif, we are looking for an angle in the interval File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image417.gif with a cosine value of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image491.gif.  The angle that satisfies this is File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image493.gif.

 

d) Evaluating File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image469.gif, we are looking for an angle in the interval File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image438.gif with a tangent value of 1.  The correct angle is File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image495.gif.

Evaluate File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image505.gif using your calculator.

 

Since the output of the inverse function is an angle, your calculator will give you a degree value if in degree mode, and a radian value if in radian mode.

 

In radian mode, File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image507.gif          In degree mode, File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image509.gif

Try it Now

2. Evaluate File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image511.gif using your calculator.

 

In Section 5.5, we worked with trigonometry on a right triangle to solve for the sides of a triangle given one side and an additional angle. Using the inverse trig functions, we can solve for the angles of a right triangle given two sides.

Example 3
Solve the triangle for the angle θ.

Since we know the hypotenuse and the side adjacent to the angle, it makes sense for us to use the cosine function.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image514.gif              Using the definition of the inverse,

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image516.gif          Evaluating

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image518.gif, or about 41.4096°

There are times when we need to compose a trigonometric function with an inverse trigonometric function.  In these cases, we can find exact values for the resulting expressions.

 

Example 4

Evaluate File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image520.gif.

 

a) Here, we can directly evaluate the inside of the composition. 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image522.gif

Now, we can evaluate the inverse function as we did earlier.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image524.gif

Try it Now

3. Evaluate File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image526.gif.

 

Example 5

Find an exact value for File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image528.gif.

 

Beginning with the inside, we can say there is some angle so File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image530.gif, which means File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image532.gif, and we are looking for File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image534.gif.  We can use the Pythagorean identity to do this.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image536.gif                        Using our known value for cosine

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image538.gif                           Solving for sine

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image540.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image542.gif

Since we know that the inverse cosine always gives an angle on the interval File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image417.gif, we know that the sine of that angle must be positive, so File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image544.gif

 

Example 6

Find an exact value for File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image546.gif.

 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image547.gifWhile we could use a similar technique as in the last example, we will demonstrate a different technique here.  From the inside, we know there is an angle so File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image549.gif.  We can envision this as the opposite and adjacent sides on a right triangle.

 

Using the Pythagorean Theorem, we can find the hypotenuse of this triangle:

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image551.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image553.gif

 

Now, we can evaluate the sine of the angle as opposite side divided by hypotenuse

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image555.gif

This gives us our desired composition

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image557.gif

Try it Now

4. Evaluate File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image559.gif

We can also find compositions involving algebraic expressions.

 

Example 7

Find a simplified expression for File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image561.gif, for File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image563.gif.

 

We know there is an angle θ so that File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image565.gif.  Using the Pythagorean Theorem,

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image536.gif                        Using our known expression for sine

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image567.gif                           Solving for cosine

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image569.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image571.gif

Since we know that the inverse sine must give an angle on the interval File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image427.gif, we can deduce that the cosine of that angle must be positive.  This gives us

 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image573.gif

Try it Now

5. Find a simplified expression for File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image575.gif, for File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image577.gif.

Important Topics of This Section

  • Inverse trig functions:  arcsine, arccosine and arctangent
  • Domain restrictions
  • Evaluating inverses using unit circle values and the calculator
  • Simplifying numerical expressions involving the inverse trig functions
  • Simplifying algebraic expressions involving the inverse trig functions

 

Try it Now Answers

  1. a) File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image579.gif   b) File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image581.gif    c) File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image020.gif   d) File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image584.gif

 

2. 1.9823 or 113.578°

 

3. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image588.gif

4. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image590.gif

5. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image592.gif

Section 6.3 Exercises

 

Evaluate the following expressions.

1. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image594.gif             2. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image596.gif             3. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image598.gif                         4. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image600.gif         

5. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image602.gif                6. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image604.gif             7. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image606.gif          8. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image608.gif

9. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image610.gif                  10. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image612.gif             11. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image614.gif           12. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image616.gif

 

Use your calculator to evaluate each expression.

13. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image511.gif          14. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image619.gif             15. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image621.gif           16. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image623.gif

 

Find the angle θ.

17. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image624.gif                  18. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image512.gif

 

Evaluate the following expressions.

19. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image626.gif                                         20. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image628.gif

21. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image630.gif                                       22. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image632.gif

23. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image634.gif                                          24. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image636.gif

25. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image638.gif                                             26. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image640.gif

 

Find a simplified expression for each of the following.

27. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image642.gif, for File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image644.gif               28. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image646.gif, for File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image648.gif  

29. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image650.gif                                            30. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image652.gif

Contributors

  • David Lippman (Pierce College)
  • Melonie Rasmussen (Pierce College)