Inverse Trig Functions
Original Function
Inverse
y = sin x y = sin-1 x y = arcsin x
y = cos x y = cos-1 x y = arccos x
y = tan x y = tan-1 x y = arctan x
REVIEW SLIDE
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
REVIEW SLIDE
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.
REVIEW SLIDE
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 = Sin-1 x y = Arcsin x
y = Cos x y = Cos-1 x y = Arccos x
y = Tan x y = Tan-1 x y = Arctan x
REVIEW SLIDE
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
REVIEW SLIDE
x
yy = sin(x)
12
2
3
3
2
2
4
2
1
6
002
1
6
2
2
4
2
3
3
12
)(
xfx
Sketch a graph of y = Sin x Remember principal values
Create a table
21
32
342
262
10062
142
232
32
1
)(sin 1
xx
Now use your table to generate Sin-1
12
2
3
3
2
2
4
2
1
6
002
1
6
2
2
4
2
3
3
12
)sin(
xx
IF YOU CAN REMEMBER AND MEMORIZE WHAT THE original and inverse funcitons look like, it will make your life a lot easier!!!
Table of Values of Cos x and Arccos xy = Cos x
X Y
0 1
π/3 0.5
π/2 0
2π/3 -0.5
π -1
y = Arccos x
X Y
1 0
0.5 π/3
0 π/2
-0.5 2π/3
-1 π
x
y
x
y
The other trig functions require similar restrictions on their domains in order to generate an inverse.
Like the sine function, the domain of the section of the
tangent that generates the arctan is .
2,
2
,
2,
2RandD
2,
2,
RandD
Y=Tan(x)Y=Arctan(x)
Table of Values of Tan x and Arctan xy = Tan x
X Y
-π/2 UD
-π/4 -1
0 0
π/4 1
π/2 UD
y = Arctan x
X Y
UD -π/2
-1 -π/4
0 0
1 π/4
UD π/2
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 = 2Tan x
X Y
-π/2 Undefined
-π/4 -2
0 0
π/4 2
π/2 Undefined
Now graph the original function, y = Arctan ½x by switching the table you just completed!
1
-1
π/2-π/2
Now graph the original function, y = Arctan ½x by switching the table you just completed!
y = 2Tan x
X Y
-π/2 UD
-π/4 -2
0 0
π/4 2
π/2 UD
y = Arctan ½x
X Y
UD -π/2
-2 -π/4
0 0
2 π/4
UD π/2 1-1
π/2
-π/2
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) = 2y½Arcsin(x) = y
Let’s graph y = Sin(2x) first.The domain changes because of the 2, how?
Now graph the inverse function, y = Arcsin(x)/2 by switching the table you just completed!
y = Sin2x
X Y
-π/4 -1
-π/12 -.5
0 0
π/12 .5
π/4 1
1
-1
π/2 π-π/2
y = ½Arcsin(x)
X Y
-1 -π/4
-.5 -π/12
0 0
.5 π/12
1 π/4Remember you can always check and see if they are symmetric with respect to y = x
€
−2
≤ 2x ≤π
2
€
−4
≤ x ≤π
4
Divide all sides by 2
Graph the inverse of:
€
y =π
2+ Arc sin x
Let’s find the inverse equation first:
€
x =π
2+ Arc sin y
Flip the “x” and “y” and solve for “y”:
€
x −π
2= Arc sin y
€
Sin x −π
2
⎛
⎝ ⎜
⎞
⎠ ⎟= y
X Y
0 -1
π/4
π/2 0
3π/4
π 1
€
y = Sin x −π
2
⎛
⎝ ⎜
⎞
⎠ ⎟
€
− 22
€
22
Take the sine of both sides
Domain is now:
€
−2
≤ x −π
2≤π
2
€
0 ≤ y ≤ π
Add π/2 to all three sides
1
-1
π/2 π
Graph
€
y =π
2+ Arc sin x
y = (π/2)+Arcsin x
X Y
-1 0
π/4
0 π/2
3π/4
1 π
Since we are graphing Arcsin the domain will become the range, but it will change!!Solve for x:
€
y =π
2+ Arc sin x
€
y −π
2= Arc sin x
€
sin y −π
2
⎛
⎝ ⎜
⎞
⎠ ⎟= x
Domain is now:
€
−2
≤ y −π
2≤π
2
€
0 ≤ y ≤ π
Add π/2 to all three sides
Take the sine of both sides
€
− 22
€
22
€
−2
≤ x ≤π
2
What is the domain for Sin(x)?
1-1
π/2
π
Now make a table using the y-values as your input into this function:
y = Arcsinx
Just shifted up π/2
Now try to graph just by using the shifting technique.
€
y = Arc sin x −π
4
1-1
π/2
π
y = Arcsinx
Just shifted down π/4
-π/2
Now try to graph just by using the shifting technique.
€
y = Arc tan x +π
4
1-1
π/2
π
y = Arctan(x)
Just shifted up π/4
-π/2
Determine if each of the following is true or false. If false give a counter example.
1. Cos-1(cos x) = x for all values of x
2. Sin-1(sin x) = x for all values of x
3. Sin-1x + Cos-1x = π/2 -1 ≤ x ≤ 1
4. Arccos x = Arccos (-x) -1 ≤ x ≤ 1
5. Tan-1x = 1/(Tan x)
x = 270°
Cos-1(cos 270°) = Cos-1(0) = 90°FALSE
FALSE
TRUE
FALSE
FALSE
x = 180° or try x = 225°
Sin-1(sin 180°) = Sin-1(0) = 0°
x = 0 of try x = 1 or -1
Sin-1(0) + Cos-1(0) = 0° + 90°= 90°
x = 1 or try x = -1
Arccos(1) ≠ Arccos (-1) 0° ≠ 180°
x = 0
Tan-1 (0) ≠ 1 / (Tan (0)) 0° ≠ 1 / 0 UNDEFINDED