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352
APPENDIX B. TRIGONOMETRY BASICS
B.4
Inverse Trigonometric Functions
Trigonometric functions are periodic functions defined in terms of the unit
circle. A function must be one-to-one in order to have an inverse function.
Periodic functions obviously fail the horizontal line test and are not one-to-one.
However, we can restrict the function to a domain on which it is one-to-one.
For functions defined in terms of the unit circle, we will restrict each domain
to include the first quadrant, corresponding to angles from 0 to π2 , as well as
adjacent angles that will guarantee the restricted function is one-to-one and
includes the full range of output values.
Recall that the sine function represents the y-coordinate of the point on
the unit circle of an angle’s terminal edge. The range consists of all numbers
in the interval [−1, 1]. The first quadrant of angles 0 to π2 leads to points on
the unit circle with y-values from 0 to 1. Angles just beyond π2 repeat the
same y-values. We instead use angles in the fourth quadrant from − π2 to 0.
Altogether, the restricted domain will be [− π2 , π2 ].
Figure B.4.1: The restricted domain for sine is the interval [− π2 , π2 ].
y = sin(θ)
y
1
0
−1
−π
− π2
0
π
2
π
2π
θ
Figure B.4.2: The sine function restricted to the domain [− π2 , π2 ].
The inverse function for the restricted sine function is called the arcsine
function or inverse sine function. Because sin takes an angle θ in radians as
the input and gives the y-coordinate on the unit circle as the output, we have
B.4. INVERSE TRIGONOMETRIC FUNCTIONS
353
sin : θ 7→ y. The inverse takes a y-coordinate on the unit circle as the input
and gives an angle θ in the interval [− π2 , π2 ] as output, we have sin−1 : y 7→ θ.
The graph of the arcsine is shown below.
θ
π
2
θ = sin−1 (y)
0
− π2
−1
0
y
1
Figure B.4.3: The arcsine function, which is the inverse of the restricted sine.
The cosine function takes an angle θ as the input and returns the xcoordinate of the corresponding point on the unit circle. The first quadrant
angles between θ = 0 and θ = π2 have x-coordinates between 0 and 1. To
obtain the x-coordinates between −1 and 0 come from angles in the second
quadrant. The restricted domain will be the interval [0, π].
Figure B.4.4: The restricted domain for cosine is the interval [0, π].
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APPENDIX B. TRIGONOMETRY BASICS
x = cos(θ)
x
1
0
−1
−π
− π2
0
π
2
π
2π
θ
Figure B.4.5: The cosine function restricted to the domain [0, π].
The inverse function for the restricted cosine function is called the arccosine
function or inverse cosine function. Because cos takes an angle θ in radians as
the input and gives the x-coordinate on the unit circle as the output, we have
cos : θ 7→ x. The inverse takes an x-coordinate on the unit circle as the input
and gives an angle θ in the interval [0, π] as output, so we have cos−1 : x 7→ θ.
The graph of the arccosine is shown below.
θ
θ = cos−1 (y)
π
2
0
−1
0
y
1
Figure B.4.6: The arccosine function, which is the inverse of the restricted
cosine.
The tangent function takes an angle θ as the input and returns the ratio xy
for the coordinates (x, y) of the corresponding point on the unit circle. This
ratio is exactly the slope m = xy of the line joining (0, 0) and (x, y). The first
quadrant angles between θ = 0 and θ = π2 correspond to all of the possible
positive slopes. To obtain negative slopes, we could use either the second or
fourth quadrant. So that the function will be continuous, the restricted domain
is chosen as the interval (− π2 , π2 ). The end-points are not included because the
tangent is not defined at those angles.
B.4. INVERSE TRIGONOMETRIC FUNCTIONS
355
Figure B.4.7: The restricted domain for tangent is the interval (− π2 , π2 ).
m
m = tan(θ)
1
0
−1
−π
− π2
0
π
2
π
2π
θ
Figure B.4.8: The tangent function restricted to the domain (− π2 , π2 ).
The inverse function for the restricted tangent function is called the arctangent function or inverse tangent function. Because tan takes an angle θ in
radians as the input and gives the slope m of the angle, tan : θ 7→ m. The
inverse takes a slope m as the input and gives an angle θ in the interval (− π2 , π2 )
the has this slope, so we have tan−1 : m 7→ θ. The graph of the arctangent is
shown below.
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APPENDIX B. TRIGONOMETRY BASICS
θ = tan−1 (m)
θ
π
2
0
− π2
−8
−6
−4
−2
0
m
2
4
6
8
Figure B.4.9: The arctangent function, which is the inverse of the restricted
tangent.
The secant, cosecant, and cotangent functions also have restricted domains
and corresponding inverse functions. The table below summarizes the restricted domains and ranges for each of the trigonometric functions.
Restricted Function
sin(x)
cos(x)
tan(x)
cot(x)
sec(x)
csc(x)
Domain
[− π2 , π2 ]
[0, π]
(− π2 , π2 )
(0, π)
[0, π2 ) ∪ ( π2 , π]
[− π2 , 0) ∪ (0, π2 ]
Range
[−1, 1]
[−1, 1]
(−∞, ∞)
(−∞, ∞)
(−∞, −1] ∪ [1, ∞)
(−∞, −1] ∪ [1, ∞)
The domain and range for the inverse functions are exactly the reverse of the
restricted trigonometric functions. The inverse trigonometric functions have
multiple representations. For example, the arcsine is sometimes written sin−1
but is also written either arcsin or asin. The table summarizes the information
about the inverse trigonometric functions.
Inverse Functions
sin−1 (x) = arcsin(x)
cos−1 (x) = arccos(x)
tan−1 (x) = arctan(x)
cot−1 (x) = arccot(x)
sec−1 (x) = arcsec(x)
csc−1 (x) = arccsc(x)
Domain
[−1, 1]
[−1, 1]
(−∞, ∞)
(−∞, ∞)
(−∞, −1] ∪ [1, ∞)
(−∞, −1] ∪ [1, ∞)
Range
[− π2 , π2 ]
[0, π]
(− π2 , π2 )
(0, π)
[0, π2 ) ∪ ( π2 , π]
[− π2 , 0) ∪ (0, π2 ]