Inverse Trigonometric Functions

(Section 18)

In addition to knowing the six trigonometric functions it is also important that a calculus student understand all six inverse trigonometric functions: Inverse Sine, Inverse Cosine, Inverse Tangent, Inverse Cotangent, Inverse Secant and Inverse Cosecant.

An inverse trigonometric function is defined from its corresponding trigonometric function.

Recall that the y = sin x has a domain of and a range of [-1,1]. To define the inverse function we first exchange the place of the x and y to yield x = sin y. This is equivalent to saying y = inverse sin x. It is written . Remember that when you say you are really stating that . So .

Since all the trigonometric functions do not pass the horizontal line test, it is necessary to restrict the domain of the original trigonometric function so an inverse function will exist.

When the x and y switch roles, this causes the domain and range switch too.

The graphs for the inverse functions will the original graph reflected over the line y = x.

y= inverse sin x y = inverse cos x y = inverse tan x in the connected mode

Domain: [-1,1] Domain:[-1,1] Domain:

Range: Range: Range:

y = inverse cot x in the connected mode y= inverse sec x in the connected mode y = inverse csc x in the connected mode

Domain: Domain: Domain:

Range: Range: Range:

To graph on the graphing calculator enter To graph on the graphing calculator enter To graph on the graphing calculator enter

It is important to memorize the range values associated with each of these inverse functions so you understand why the calculator is giving you a particular answer. If you want an answer in another interval, you will have to make adjustments based upon your knowledge of the six basic trigonometric functions.

Suppose you want to find the solution to the equation for . You will recall from the above range values that the calculator will only give you an answer for this equation between . The answer the calculator will give is . What do you know about the graph of the sine function which could help you answer this question. The answer we are looking for is in quadrant II and from . This would make the answer .