6.8 – Trig Inverses and their graphs

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6.8 – TRIG INVERSES AND THEIR GRAPHS
Quick Review
How do you find inverses of functions?
Are inverses of functions always functions? How did we test for this?
Inverse Trig Functions
Original Function
Inverse
y = sin x y = sin1 x y = arcsin x
y = cos x y = cos1 x y = arccos x
y = tan x y = tan1 x y = arctan x
Consider the graph of y = sin x
What is the domain and range of sin x?
What would the graph of y = arcsin x look like?
What is the domain and range of arcsin x?
Domain: all real numbersRange: [1, 1]
Domain: [1, 1]Range: all real numbers
Is the inverse of sin x a function? This will also be true for
cosine and tangent. Therefore all of the
domains are restricted in order for the inverses to be functions.
How do you know if the domain is restricted for the original functions? Capital letters are used to distinguish
when the function’s domain is restricted.
Original Functions with
Restricted Domain
Inverse Function
y = Sin x y = Sin1 x y = Arcsin x
y = Cos x y = Cos1 x y = Arccos x
y = Tan x y = Tan1 x y = Arctan x
Original Domains Restricted Domains
Domain Range
y = sin x
all real numbers
y = Sin x y = sin x y = Sin x
y = cos xall real
numbers
y = Cos x y = cos x y = Cos x
y = tan xall real
numbers except n,
where n is an odd integer
y = Tan x y = tan x
all real numbers
y = Tan x
all real numbers
Complete the following table on your own
Function Domain Range
y = Sin x
y = Arcsin x
y = Cos x
y = Arccos x
y = Tan xall real numbers
y = Arctan x
Table of Values of Sin x and Arcsin x
y = Sin x
X Y
π/2
π/6
0
π/6
π/2
y = Arcsin x
X Y
π/2
π/6
0
π/6
π/2
Why are we using these values?
Graphs of Sin x and Arcsin x
Table of Values of Cos x and Arccos x
y = Cos x
X Y
0
π/3
π/2
2π/3
π
y = Arccos x
X Y
0
π/3
π/2
2π/3
π
Why are we using these values?
Graphs of Cos x and Arccos x
Table of Values of Tan x and Arctan x
y = Tan x
X Y
π/2
π/4
0
π/4
π/2
y = Arctan x
X Y
π/2
π/4
0
π/4
π/2
Why are we using these values?
Graphs of Tan x and Arctan x
Write an equation for the inverse of y = Arctan ½x. Then graph the function and its inverse.
To write the equation:1.Exchange x and y2.Solve for y
x = Arctan ½yTan x = ½y2Tan x = y
Let’s graph 2Tan x = y first.Complete the table:
Then graph!
y = Tan x
X Y
π/2
π/4
0
π/4
π/2Now graph the original function, y = Arctan ½x by switching the table you just completed!
Write an equation for the inverse of y = Sin(2x).
Then graph the function and its inverse.
To write the equation:1.Exchange x and y2.Solve for y
x = Sin(2y)Arcsin(x) = 2yArcsin(x)/2 = y
Let’s graph y = Sin(2x) first.Why are these xvalues used?
Now graph the inverse function, y = Arcsin(x)/2 by switching the table you just completed!
y = Sin2x
X Y
π/4
π/12
0
π/12
π/4
Evaluate each expression
Evaluate each expression