6
6.1 The Inverse Trig Functions
REVIEW: Properties of Inverse Functions
▪ [pic] for every x in the domain of f and [pic] for every x in the domain of [pic].
▪ Domain of f = range of [pic], and range of f = domain of [pic] (switch the x and y values!)
▪ The graph of f and the graph of [pic] are symmetric with respect to the line [pic].
▪ If f(x) is one-to-one (passes horizontal line test), then its inverse is a function.
▪ If a function [pic] has an inverse function, the equation of the inverse function is [pic]. The solution of this equation is [pic].
** If a function is not one-to-one (it does not pass the horizontal line test) it may be possible to restrict its domain so that the restricted function is one to one! Do this to create an inverse that IS a function (
Sine Function
y = sin(x)
Restricted Domain: ________
Range: ________
Cosine Function
y = cos(x)
Restricted Domain: ________
Range: ________
Tangent Function
y = tan(x)
Restricted Domain: ________
Range: ________
The table below switches the Domain and Range of the original (sine, cosine, tangent) functions in order to describe their Inverse Trig Functions.
|Inverse Trig | Domain | Range |
|Function |(x-values) |(angles in rad) |
|[pic] | | |
|arcsin(x) | | |
|[pic] | | |
|arcos(x) | | |
|[pic] | | |
|arctan(x) | | |
You MUST memorize…
Examples: (always give radian value unless specifically told otherwise()
#1] Find the exact value of the following:
a) [pic]= ? b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
#2] Use a calculator to find an approximate value of the following:
a) [pic]= ? b) [pic] = ?
c) [pic] = ? d) [pic]
|[pic] where [pic] |
| |
|[pic] where [pic] |
#3] Evaluate the expression without a calculator.
a) [pic] b) [pic]= ?
| |
|[pic] where [pic] |
| |
|[pic] where [pic] |
#4] Evaluate the expression without a calculator.
a) [pic] b) [pic]= ?
|[pic] where [pic] |
| |
|[pic] where [pic] |
#5] Evaluate the expression without a calculator.
a) [pic] b) [pic]= ?
HW: pg. 423 #13 – 43 odd, #45 – 56 ALL
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