Learning Outcomes
- Evaluate inverse trigonometric functions
The six basic trigonometric functions are periodic, and therefore they are not one-to-one. However, if we restrict the domain of a trigonometric function to an interval where it is one-to-one, we can define its inverse. Consider the sine function. The sine function is one-to-one on an infinite number of intervals, but the standard convention is to restrict the domain to the interval [−π2,π2][−π2,π2]. By doing so, we define the inverse sine function on the domain [−1,1][−1,1] such that for any xx in the interval [−1,1][−1,1], the inverse sine function tells us which angle θθ in the interval [−π2,π2][−π2,π2] satisfies sinθ=xsinθ=x. Similarly, we can restrict the domains of the other trigonometric functions to define inverse trigonometric functions, which are functions that tell us which angle in a certain interval has a specified trigonometric value.
Definition
The inverse sine function, denoted sin−1sin−1 or arcsin, and the inverse cosine function, denoted cos−1cos−1 or arccos, are defined on the domain D={x|−1≤x≤1}D={x|−1≤x≤1} as follows:
The inverse tangent function, denoted tan−1tan−1 or arctan, and inverse cotangent function, denoted cot−1cot−1 or arccot, are defined on the domain [latex]D=\{x|-\infty
The inverse cosecant function, denoted csc−1csc−1 or arccsc, and inverse secant function, denoted sec−1sec−1 or arcsec, are defined on the domain D={x||x|≥1}D={x||x|≥1} as follows:
To graph the inverse trigonometric functions, we use the graphs of the trigonometric functions restricted to the domains defined earlier and reflect the graphs about the line y=xy=x (Figure 16).

Figure 16. The graph of each of the inverse trigonometric functions is a reflection about the line y=xy=x of the corresponding restricted trigonometric function.
When evaluating an inverse trigonometric function, the output is an angle. For example, to evaluate cos−1(12)cos−1(12), we need to find an angle θθ such that cosθ=12cosθ=12. Clearly, many angles have this property. However, given the definition of cos−1cos−1, we need the angle θθ that not only solves this equation, but also lies in the interval [0,π][0,π]. We conclude that cos−1(12)=π3cos−1(12)=π3. Review the following table of common sine and cosine values using reference angles, if necessary.
Recall: Commonly encountered angles in first quadrant of the unit circle
Angle | 0 | π6π6, or 30° | π4π4, or 45° | π3π3, or 60° | π2π2, or 90° |
Cosine | 1 | √32√32 | √22√22 | 1212 | 0 |
Sine | 0 | 1212 | √22√22 | √32√32 | 1 |
We now consider a composition of a trigonometric function and its inverse. For example, consider the two expressions sin(sin−1(√22))sin(sin−1(√22)) and sin−1(sin(π))sin−1(sin(π)). For the first one, we simplify as follows:
For the second one, we have
The inverse function is supposed to “undo” the original function, so why isn’t sin−1(sin(π))=πsin−1(sin(π))=π? Recalling our definition of inverse functions, a function ff and its inverse f−1f−1 satisfy the conditions f(f−1(y))=yf(f−1(y))=y for all yy in the domain of f−1f−1 and f−1(f(x))=xf−1(f(x))=x for all xx in the domain of ff, so what happened here? The issue is that the inverse sine function, sin−1sin−1, is the inverse of the restricted sine function defined on the domain [−π2,π2][−π2,π2]. Therefore, for xx in the interval [−π2,π2][−π2,π2], it is true that sin−1(sinx)=xsin−1(sinx)=x. However, for values of xx outside this interval, the equation does not hold, even though sin−1(sinx)sin−1(sinx) is defined for all real numbers xx.
What about sin(sin−1y)sin(sin−1y)? Does that have a similar issue? The answer is no. Since the domain of sin−1sin−1 is the interval [−1,1][−1,1], we conclude that sin(sin−1y)=ysin(sin−1y)=y if −1≤y≤1−1≤y≤1 and the expression is not defined for other values of yy. To summarize,
and
Similarly, for the cosine function,
and
Similar properties hold for the other trigonometric functions and their inverses.
Example: Evaluating Expressions Involving Inverse Trigonometric Functions
Evaluate each of the following expressions.
- sin−1(−√32)sin−1(−√32)
- tan(tan−1(−1√3))tan(tan−1(−1√3))
- cos−1(cos(5π4))cos−1(cos(5π4))
- sin−1(cos(2π3))sin−1(cos(2π3))
Watch the following video to see the worked solution to Example: Evaluating Expressions Involving Inverse Trigonometric Functions
Activity: The Maximum Value of a Function
In many areas of science, engineering, and mathematics, it is useful to know the maximum value a function can obtain, even if we don’t know its exact value at a given instant. For instance, if we have a function describing the strength of a roof beam, we would want to know the maximum weight the beam can support without breaking. If we have a function that describes the speed of a train, we would want to know its maximum speed before it jumps off the rails. Safe design often depends on knowing maximum values.
This project describes a simple example of a function with a maximum value that depends on two equation coefficients. We will see that maximum values can depend on several factors other than the independent variable xx.
- Consider the graph in Figure 17 of the function y=sinx+cosxy=sinx+cosx. Describe its overall shape. Is it periodic? How do you know?
Figure 17. The graph of y=sinx+cosxy=sinx+cosx.
Using a graphing calculator or other graphing device, estimate the xx– and yy-values of the maximum point for the graph (the first such point where x>0x>0). It may be helpful to express the xx-value as a multiple of ππ.
- Now consider other graphs of the form y=Asinx+Bcosxy=Asinx+Bcosx for various values of AA and BB. Sketch the graph when A=2A=2 and B=1B=1, and find the xx– and yy-values for the maximum point. (Remember to express the xx-value as a multiple of ππ, if possible.) Has it moved?
- Repeat for A=1,B=2A=1,B=2. Is there any relationship to what you found in part (2)?
- Complete the following table, adding a few choices of your own for AA and BB:
AA BB xx yy AA BB xx yy 0 1 √3√3 1 1 0 1 √3√3 1 1 12 5 1 2 5 12 2 1 2 2 3 4 4 3 - Try to figure out the formula for the yy-values.
- The formula for the xx-values is a little harder. The most helpful points from the table are (1,1),(1,√3),(√3,1)(1,1),(1,√3),(√3,1). (Hint: Consider inverse trigonometric functions.)
- If you found formulas for parts (5) and (6), show that they work together. That is, substitute the xx-value formula you found into y=Asinx+Bcosx and simplify it to arrive at the y-value formula you found.
Try It
Candela Citations
- 1.4 Inverse Functions. Authored by: Ryan Melton. License: CC BY: Attribution
- Calculus Volume 1. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/details/books/calculus-volume-1. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-1/pages/1-introduction