Background
A function has an inverse function if it is 1-1.
A function is 1-1 if it passes the horizontal line test.
Since the trigonometric functions are periodic, none of them are 1-1.
If the domain is restricted, each trigonometric function can be 1-1.
There are an infinite number of choices for a restricted domain.
However, there are standard restricted domains for each function.
The Inverse Sine
The inverse sine function, denoted f(x) = sin-1x or f(x) = arcsinx, is the inverse of g(x) = sinx.
Like sinx, sin-1x and arcsinx are not sin-1 and arcsin times x but are the inverse sine function
evaluated at x.
Also, sin-1x is not 1/sinx.
I personally prefer and use f(x) = sin-1x.
To define f(x) = sin-1x
,
.
2 2
On this interval, the range of g(x) = sinx is still [-1, 1].
The graph of g(x) = sinx on its restricted domain.
To make g(x) = sinx 1-1, restrict the domain to
Reflect around y = x.
Therefore, f(x) = sin-1x
Domain: [-1, 1]
Range:
2
,
2
The graph of f(x) = sin-1x
The Inverse Cosine
The inverse cosine function, denoted f(x) = cos-1x or f(x) = arccosx, is the inverse of g(x) = cosx.
To define f(x) = cos-1x
To make g(x) = cosx 1-1, restrict the domain to [0, π].
On this interval, the range of g(x) = cosx is still [-1, 1].
The graph of g(x) = cosx on its restricted domain.
Reflect around y = x.
Therefore, f(x) = cos-1x
Domain: [-1, 1]
Range: [0, π]
The graph of f(x) = cos-1x
The Inverse Tangent
The inverse tangent function, denoted f(x) = tan-1x or f(x) = arctanx, is the inverse of g(x) = tanx.
To define f(x) = tan-1x
,
.
2 2
On this interval, the range of g(x) = sinx is still (-∞, ∞).
The graph of g(x) = tanx on its restricted domain.
To make g(x) = tanx 1-1, restrict the domain to
Reflect around y = x.
Therefore, f(x) = tan-1x
Domain: (-∞, ∞)
,
2 2
The graph of f(x) = tan-1x
Range:
Examples
Find sin-1(-1), sin-1(-1/2), sin-1(0), sin
Find cos
3
1
2
2
, and sin-1(7/8).
2
1
, cos-1(0), cos-1(1/2), cos-1(1), and cos-1(-0.147).
Find tan-1(- 3 ), tan-1(-1), tan-1(0), tan
1
3
, and tan-1(7.8).
3
Properties
sin(sin-1x) = x for x in [-1, 1] and sin-1(sinx) = x for x in
,
.
2 2
cos(cos-1x) = x for x in [-1, 1] and cos-1(cosx) = x for x in [0, π].
tan(tan-1x) = x for x in (-∞, ∞) and tan-1(tanx) = x for x in
2
,
2
.
Uses f(f-1(x)) = x and f-1(f(x)) = x.
Examples
sin(sin-1(-1/2))
sin(sin-1(-0.75))
sin-1(sin(-π/6))
sin-1(sin(2π/3))
cos-1(cos(π/4))
tan(tan-1(1))
cos-1(cos(-0.329))
tan-1(tan(3π/4))
cos(cos-1(- 3 /2))
tan-1(tan(π/6))
cos(cos-1(0.7))
tan(tan-1(2.91))
The Other Inverse Functions
The inverse cosecant function, denoted by f(x) = csc-1x = arccscx, is the inverse of g(x) = cscx.
f(x) = csc-1x = sin-1(1/x)
Rationale
Let y = csc-1x.
By definition of the inverse, cscy = x.
Using the definition of cscy, 1/siny = x.
Solving for siny, siny = 1/x.
By definition of the inverse, y = sin-1(1/x)
The inverse secant function, denoted by f(x) = sec-1x = arcsecx, is the inverse of g(x) = secx.
f(x) = sec-1x = cos-1(1/x)
The inverse cotangent function, denoted by f(x) = cot-1x = arccotx, is the inverse of g(x) = cotx.
f(x) = cot-1x = tan-1(1/x)
If tan-1(1/x) is negative, add π, because cot-1x has a range is (0, π).