Definition of Inverse Trigonometric Functions

[nycmathguy]

Section 4.7

Can you please give an example for each case?

 

[MathLover1]

definition of the inverse trigonometric function
Inverse trigonometric functions are simply defined as the inverse functions of the basic trigonometric functions which are sine, cosine, tangent, cotangent, secant, and cosecant functions. They are also termed as arcus functions, antitrigonometric functions or cyclometric functions.

y=arcsin(x) if and only if sin(y)=x domain: -1<=x<=1 range -pi/2<=y<=pi/2

f and f^-1 is the inverse
• The range of f = the domain of f^-1, the inverse.
• The domain of f = the range of f ^-1 the inverse.

Trigonometric functions are periodic, therefore each range value is within the limitless
domain values (no breaks in between).

Since trigonometric functions have no restrictions, there is no inverse.

• With that in mind, in order to have an inverse function for trigonometry, we restrict the
domain of each function, so that it is one to one.

• A restricted domain gives an inverse function because the graph is one to one and able to pass
the horizontal line test.

How to restrict a domain:
– Restrict the domain of the sine function, y = sin (x), so that it is one to one, and not infinite
by setting an interval [-π/2, π/2]

The restricted sine function passes the horizontal line test, therefore it is one to one
– Each range value (-1 to 1) is within the limited domain (-π/2, π/2).

Inverse Sine Function
• sin^-1 or arcsin is the inverse of the restricted sine function, y = sin (x), [-π/2, π/2]

The equations -> y = sin^-1(x) or y = arcsin (x) which also means, sin (y) = x, where -π/2 < y < π/2, -1 < x < 1 (remember f range is f ^-1 domain and vice versa).

 

[nycmathguy]

definition of the inverse trigonometric function
Inverse trigonometric functions are simply defined as the inverse functions of the basic trigonometric functions which are sine, cosine, tangent, cotangent, secant, and cosecant functions. They are also termed as arcus functions, antitrigonometric functions or cyclometric functions.

y=arcsin(x) if and only if sin(y)=x domain: -1<=x<=1 range -pi/2<=y<=pi/2

f and f^-1 is the inverse
• The range of f = the domain of f^-1, the inverse.
• The domain of f = the range of f ^-1 the inverse.

Trigonometric functions are periodic, therefore each range value is within the limitless
domain values (no breaks in between).
View attachment 1143


Since trigonometric functions have no restrictions, there is no inverse.

• With that in mind, in order to have an inverse function for trigonometry, we restrict the
domain of each function, so that it is one to one.

• A restricted domain gives an inverse function because the graph is one to one and able to pass
the horizontal line test.

How to restrict a domain:
– Restrict the domain of the sine function, y = sin (x), so that it is one to one, and not infinite
by setting an interval [-π/2, π/2]

View attachment 1144

The restricted sine function passes the horizontal line test, therefore it is one to one
– Each range value (-1 to 1) is within the limited domain (-π/2, π/2).

Inverse Sine Function
• sin^-1 or arcsin is the inverse of the restricted sine function, y = sin (x), [-π/2, π/2]

The equations -> y = sin^-1(x) or y = arcsin (x) which also means, sin (y) = x, where -π/2 < y < π/2, -1 < x < 1 (remember f range is f ^-1 domain and vice versa).

Excellent study notes. I will keep this material forever.