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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3.2 Logarithmic Functions and Their Graphs

Understanding Logarithmic Functions

Definition of a logarithmic function: the inverse of an exponential function where a is the base

where the anti exponential function is equal to the logarithmic function

Note: Exponential function have x-inputs, while the logarithmic functions have x-outputs

Evaluating Logarithmic Functions:

Example #1: 

Write the logarithmic equation in exponential form 

 

Solution: 2 is the base of the equation, or a. Therefore according to the definition of a logarithmic function, the exponential form is:

where y=8, x=3, and a=2

Example #2:

Solve the equation for x

Solution: put the logarithmic equation in exponential form 

then simplify the equation so that you have a common base

in order to solve for x, set the exponents equal to one another, such as:

and solve for x

Final Answer: 

Example #3

Solve the equation for x

Solution: put the equation in exponential form

Final Answer: Impossible, because there is not an exponent that makes 3 equal -81

Calculator Tips

There are two functions that are used so commonly, they have made functions on the calculator making solving the equations faster and more efficient

Common log- 

Natural log-

Graph of a log function

Remember that a log function is the inverse of an exponential function, so it's the graph of a logarithmic equation is a reflection of the exponential equation over the line y=x

the red line graph represents the log function, as you can see it is a reflection over the line y=x

Exponential Function Properties:

D: 

R:

x-intercept: none 

y-intercept: (0,1)

Vertical Asymptote: none

Horizontal Asymptote: y=0

Logarithmic Function Properties:

D:

R:

x-intercept: (1,0)

y-intercept: none

Vertical Asymptote: x=0

Horizontal Asymptote: none

*NOTE: the domain and range, as well as the intercepts and asymptotes, are inverses of the exponential function(they switch)

Shifting of a logarithmic graph

d=shift vertical(upwards and downwards)

c=shift horizontally(right and left)

a=vertical stretch/compression, if (-) it is reflected over the x-axis

b= when b is larger, it causes the incline to happen slower, rate of change