5.5 – Inverse Trigonometric Functions and their Graphs
5.5 Video 1 of 3:Inverse Sine Function
5.5 Video 2 of 3:Inverse Cosine Function
5.5 Video 3 of 3:Inverse Tangent Function
Section Notes
Inverse Trigonometric Functions and their Graphs
Recall that the inverse function is a function which “undoes” the function . That is, the composite of a function and its inverse results in . Mathematically, that is . For a function to have an inverse, it is important that it is a onetoone function. None of the six trigonometric functions are onetoone, so they can not have an inverse function. However, if the domain of a trigonometric function is strategically limited, then we can create a onetoone function on that domain. Such a function would have an inverse.
Sine
We saw previously that the sine function is periodic, and continues forever in both the positive and negative direction:
But if we choose a domain in such a way that no value output appears twice, we are limited to half of a period of the sine function, selected in blue:
Thus, we will consider the function on the domain :
The graph of the inverse sine function is shown here:
The Inverse Sine Function 

On the domain , the sine function has an inverse, noted either or . The domain of is and its range is , is defined by And holds the properties

To evaluate the inverse sine function, it is often easier to convert it into a sine equation, and solve:
Find the value of 

We can start by writing an equation. The expression is equal to an unknown value, so we can write Which is equivalent to the sine equation Referring to the unit circle if necessary, we can see that for values such as , but only one of these values lies between , so we have a unique solution 
Evaluate the expression 

Proceeding similarly to the previous, we have: Thus, the only value between for which the sine value is is . 
Evaluate 

Again, paying attention to the range of we have 
Evaluate the expression 

This expression has no value, as is outside the domain of . This can further be illustrated by rewriting the expression as a equation: But there are no such values, since has a range of . 
Similar to the function, there are values of that require the use of a calculator to evaluate. Many calculators have a , while others have an button, and others still might have an inverse button that, when pressed, triggers the inverse of the next button pushed. It is important to know how to use your calculator for this function.
Use a calculator to evaluate . Round your answer to four decimal places. 


Use a calculator to evaluate 


Some expressions have both a and a in them. They key is to be aware of the domains involved!
Evaluate the exrpession 

Because is within the interval , then the and cancel each other, and we have 
Evaluate the expression 

First note that is not in the interval , so the and functions do not simply cancel each other out. But note that , so we can substitute this value into the expression: rewriting the expression as a expression gives: The value of within the interval that solves this equation is , thus, 
Such expressions can be simplified by using the reference angles that lie within the interval :
Evaluate the expression 


Cosine
Similar to the sine function, the cosine function is periodic:
Again, we can select a halfperiod of cosine which will make the function onetoone, however, this interval is not the same as the sine. We can see the halfperiod blow, highlighted in blue.
Thus, we consider the function on the domain :
The graph of the inverse cosine function is graphed here:
The Inverse Cosine Function 

On the domain , the cosine function has an inverse, noted either or . The domain of is , and has range , is defined by And holds the properties

Evaluating the cosine function can be similar to the inverse sine functions:
Find the value of 

And the value between that satisfies this equation is 
Evaluate 

So 
Evaluate 

1.5 is outside the interval ,so there is no solution for this expression. 
And expressions using both and are treated similarly to ones with s
Evaluate 

Because is outside the interval , we can replace the cosine expression: Thus, we see that the value in the interval that satisfies this equation is . 
Evaluate the expression 

Because is outside the interval , we can simplify the inside cosine expression: But is still outside the interval , so simplifying to an angle within the interval using the fact that gives: Thus, 
Tangent
Tangent has a very natural domain restriction between the two asymptotes that creates a onetoone function. Thus, we will restrict tangent to the interval in order to consider its inverse function.
A period of tangent is graphed here:
Note that the range of the tangent function is . Thus, the domain of the inverse tangent function is also . The graph of the inverse tangent is graphed here:
 
The Inverse Tangent Function 

On the domain , the tangent function has an inverse, noted either or . The domain of is all real numbers, and its range is , is defined by
And holds the properties

Evaluate 

The tangent equation on the right, of course, has solution 
Evaluate 

and the value in the interval that solves this is . 
Use a calculator to evaluate the expression . Round your answer to 4 decimal places 

