4.7 Inverse Trigonmetric Functions

Section 4.7 explains inverse trigonometric functions as well as compositions of functions.  It is convenient for trigonometric functions to have an inverse because it makes solving for angles of a right triangle much easier, but trig functions are not one-to-one.  By restricting the domain of the trig functions, each function has an inverse.

Inverse Sine

For the y = sin x graph above, the shaded area represents a section of the graph where sin x is one-to-one and the entire range is represented.  The domain [ -π/2, π/2] is where sin x has an inverse function y = arcsin x, or y = sin^-1 x.

The above graph shows y = arcsin x.  Note that the domain and range have switched from the sin x to the arcsin x graphs.  Like all inverse functions, the input- x, and the output- y, switch.  

Inverse Cosine and Inverse Tangent

Inverse Cosine

Similar to y = sin x, y = cos x and y = tan x are not one-to-one unless the domain of these functions are restricted.

For the y = cos x graph the restricted domain is [ 0, π ].  When the x and y values are switched the y =  arccos x graph looks like:

Inverse Tangent

The domain of the y = tan x graph is (-π/2, π/2) to be one-to-one.  Note that unlike the restricted domain of y = sin x with brackets, the restricted domain of y = tan x has parenthesis because y = -π/2 and y = π/2 are vertical asymptotes.

Graph of y = tan x

Graph of y = arctan x

Compositions of Functions

When dealing with an inverse trig function composed of a trig function, or a trig function composed of an inverse trig function, the most important thing to remember is that an inverse trigonometric function equals an angle measurement.  

For example, to solve-

The first step is to recognize that

 

Next, cos x = adjacent/hypotenuse, so 3 is the measure of the adjacent side and 5 is the hypotenuse.  Using the Pythagorean Theorem or special right triangles, the opposite side is found to be 4.  Finally, sin x = opposite/hypotenuse, so 4/5 is the answer.

To sum it up,

Extra

For help with the range of inverse trigonometric functions (the restricted domain of trigonometric functions), here's a table-

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