In this explainer, we will learn how to calculate exact values of trigonometric inverses and evaluate compositions of trigonometric and inverse trigonometric functions at standard angles in radians.
Let us begin by recalling some key definitions and notations related to inverse functions in general. Then, we will get into the specifics of how to evaluate inverse trigonometric functions.
Definition: Invertible Function and Related Concepts
A functionΒ maps an input belonging to the domainΒ to an output belonging to the range .
The domainΒ of the function is the set of all possible inputs such that is defined.
The range of is the set of all outputs we can get from applying to elements of .
A function is invertible if it is a one-to-one and onto function; that is, every input has one unique output and every element of the range can be written in the form for some in the domain .
Let be an invertible function. The inverse of is the function with the property
Simply stated, the inverse of a function βreversesβ the original function. We note that a function must be invertible in order to have an inverse function . If the function is invertible, then an input will map to a unique output and will map back to the original . This applies to every element in the domain and every element in the range .
As a result, the domain and range of the inverse function are essentially swapped around, compared to the original function. The range of equals the domain of and the range of equals the domain of . If the domain of does not equal the range of , we may be able to restrict the original domain to ensure they match.
Let us recall the definition of a composite function. Given two functions and , we compute the composite function by replacing each instance of in by . If two functions are inverses, their compositions will have very predictable results, according to the following rule. As we will demonstrate in this explainer, these results hold true for trigonometric functions and their inverses as well.
Rule: Composition of Inverse Functions
Let and be inverse functions. Then, applying to any in , followed by , gives us back . Equally, if we apply to any in , followed by , we get back . That is, and
Now that we have reviewed the foundational ideas about inverse functions, we will go over the special names for inverse trigonometric functions.
Key Terms: Arcsine, Arccosine, and Arctangent
The inverse function of sine is called arcsine.
The inverse function of cosine is called arccosine.
The inverse function of tangent is called arctangent.
We recall that trigonometric functions are not one to one. This means that sine, cosine, and tangent are not invertible unless we limit their domains to ensure each value of returns exactly one value of . With appropriate restrictions on , we define the three inverse trigonometric functions as follows.
Definition: Domain and Range of Inverse Trigonometric Functions
For and , we have if and only if .
For and , we have if and only if .
For and , we have if and only if .
Now, we recall that when an angle , in standard position, passes through a coordinate point , we can evaluate all six trigonometric functions using , , and according to the following definitions.
Definition: Trigonometric Functions in terms of (π₯, π¦)
When possible, it is convenient to use coordinate points from the unit circle, where . The standard angles, in radians, found on the unit circle are multiples of and between 0 and . We can convert between degrees and radians, as needed, using the fact that .
Now that we have reviewed how to evaluate standard trigonometric functions around the unit circle, we will evaluate our first inverse trigonometric function. This will require thinking βin reverseβ from a given standard trigonometric value to identify the corresponding angle , where the value is found on the unit circle. One of the challenges will be to keep in mind our newly discovered domain and range restrictions for inverse trigonometric functions.
Example 1: Evaluating Inverse Trigonometric Functions in Radians
Evaluate the expression .
Answer
We begin by recalling that arcsine is the inverse of the trigonometric function sine. In particular, gives us the angle in standard position, for which . The range of the function is in radians. This means our angle must lie somewhere within the first or fourth quadrant on the unit circle. According to the familiar CAST diagram, we know that only cosine is positive in the fourth quadrant. Since we wish to evaluate the inverse of sine at a positive value, , we can be sure that is in the first quadrant.
Now, we recall the definitions of trigonometric functions, expressed in terms of - and -coordinates from the unit circle:
To evaluate , we need to find the standard angle for which the sine of the angle is equal to .
Since , we are looking for the angle that intersects the unit circle at a -coordinate of . We already know it should be in the first quadrant, so we look there. We find the point , at .
Since , with , we conclude that .
In some questions, we may be asked to evaluate the sum or difference of inverse trigonometric functions. We will look at an example of this type next.
Example 2: Evaluating the Difference of Inverse Trigonometric Functions in Radians
Evaluate the expression .
Answer
First, we recall that arctangent is the inverse of tangent and arcsine is the inverse of sine. To approach this problem, we will evaluate and separately, then take their difference. We will proceed using radians, but we can always change angle measures to degrees if desired.
We begin by recalling that gives us the angle , in standard position, if and only if . The range of the function is .
We need to find the standard angle for which the tangent of the angle is equal to 1.
In terms of - and -coordinates from the unit circle, we define tangent as . So, we are looking for the angle that intersects the unit circle at a point , where
By multiplying each side of the above equation by , we have
Then, we simplify, resulting in the equation
This means we are looking for the point where the - and -coordinates are equal.
This only happens at two locations on the unit circle, and as shown below. In radians, these angle measures are and .
Since the range of is , we conclude that cannot be . Thus, because this angle is in the proper range and the - and -coordinates are equal.
Next, we need to evaluate . In other words, we need to find the standard angle for which the sine of the angle is equal to .
Since on the unit circle, we are looking for the angle that intersects the unit circle at a -coordinate of . The only place where we have this -coordinate on the unit circle is at the quadrantal angle .
We see from the unit circle that ; however, arcsine has a range of , so we need to find an angle coterminal with in that interval. To find a coterminal angle in radians, we add or subtract multiples of . Since is above the range of arcsine, we must subtract . Therefore, writing both terms over a common denominator of 2, we get
Therefore, .
Finally, we evaluate by substituting the values obtained above:
Subtracting a negative is equivalent to adding a positive. Therefore, by using a common denominator of 4, we have
In conclusion, we have shown that .
Now let us consider how to evaluate a composition of standard and inverse trigonometric functions.
Example 3: Evaluating a Composite Trigonometric and Inverse Trigonometric Function
Find the exact value of without using a calculator.
Answer
To answer this question, we must first recognize that the given sine function contains arccosine, which is the inverse cosine function. We call this a composite function. Let us recall the meaning of a composition of two functions. Given two functions and , we compute the composite function by replacing each instance of in by . Another way to think of this is that we substitute for the innermost values first and then work outward.
In this case, we let and ; we have to evaluate at . We begin with evaluating the innermost function, , at . This means we need to find the standard angle for which cosine equals 0.
We recall that, in terms of coordinates from the unit circle, . According to the definition of arccosine, if and only if , where . Within this range on the unit circle, we see that at . Therefore, and .
To evaluate , we substitute in the value of :
Since , then
We recall the definition of sine in terms of coordinates from the unit circle:
Therefore, we use the -coordinate of the point to evaluate :
In conclusion, we have shown that .
Let us try another example involving the composition of functions. We will use arccosine again, but this time, it is the outer function of the composition.
Example 4: Evaluating a Composite Trigonometric and Inverse Trigonometric Function
Find the exact value of without using a calculator.
Answer
To evaluate this expression, we must first recognize it as a composition of the functions arccosine (the inverse of cosine) and sine. Let us recall the meaning of a composition of two functions. Given two functions and , we compute the composite function by replacing each instance of in by .
In this case, we let and . To evaluate the composite function at , we first need to evaluate .
We recall that sine is defined as on the unit circle. So, we are looking for the -coordinate of the point where the angle intersects the unit circle. The unit circle does not display negative angle measures, so we must add radians to find the smallest positive angle coterminal with . We use a common denominator of 6 to find the coterminal angle:
We see that is in the fourth quadrant of the unit circle. At , we find the -coordinate . Therefore, .
Now, we evaluate at . Considering and , we have
Therefore,
Since , it follows that
Finally, to evaluate , we refer to the unit circle. We recall that arccosine is the inverse of cosine and that gives us the angle in standard position, for which . The definition of cosine expressed in terms of coordinates from the unit circle is . This means we are looking for the standard angle with an -coordinate of .
Let us consider what else we know about the angle we want to find. We recall that the range of the function arccosine is . Thus, our angle is somewhere within the first or second quadrant on the unit circle. According to the CAST diagram, we know that all trigonometric functions are positive in the first quadrant. We wish to evaluate the inverse of cosine at a negative value, . So, we can be sure that is in the second quadrant.
In the second quadrant, we find the -coordinate at . This means that .
In conclusion, we have shown that .
In our last example, we will explore the relationship between cosine and its inverse for various angle measures around the unit circle.
Example 5: Rewriting an Equation Using a Trigonometric Inverse
Let , where and is in radians. Find the first two positive values of , in terms of , such that .
Answer
To begin, we recall the meaning of arccosine. Arccosine is the inverse of cosine. If , then it must also be true that .
By definition, has range . This indicates is in the first or second quadrant. However, we are told that , and according to the CAST diagram, only the first or fourth quadrant has positive cosine values. Therefore, is in the first quadrant; so, the first positive value of is .
To find the next positive value of , we need to find an angle that intersects the unit circle at the same -value as . As highlighted in the diagram below, the -coordinates in the first quadrant correspond to the -coordinates in the fourth quadrant.
To find the next positive angle where cosine has the same value, we subtract from . This means that .
For example, both and are equal to because . Similarly, both and are equal to because .
In conclusion, we have shown that the required first two positive values of are and .
Let us finish by recapping some important points from the explainer.
Key Points
- The inverse trigonometric functions arcsine, arccosine, and arctangent are defined in terms of the standard trigonometric
functions, as follows:
- The inverse function of sine is called arcsine.
- For and , .
- The inverse function of cosine is called arccosine.
- For and , .
- The inverse function of tangent is called arctangent.
- For and , .
- The inverse function of sine is called arcsine.
- The unit circle is a circle with a radius of 1 whose center lies at the origin of a coordinate plane. The standard angles, in radians, found on the unit circle are multiples of and between 0 and . Each angle intersects the unit circle at a coordinate point , which helps us determine trigonometric functions at those angles.
- Trigonometric functions can be defined in terms of the - and -coordinates shown on the unit circle, as follows: This means that we look at the -coordinates to evaluate arcsine, the -coordinates to evaluate arccosine, and the quotient of corresponding - and -coordinates to evaluate arctangent.