Inverse Trigonometric Functions

Introduction

The inverse trigonometric functions answer the question: given a ratio, what angle produces it? If sin(θ)=x what is θ These functions are essential for solving equations involving trigonometric functions and appear throughout calculus in integration formulas.

Since trigonometric functions are periodic, they are not one-to-one on their natural domains. To define inverse functions, we must restrict the domains appropriately. The choice of restricted domain determines the range of the inverse function.

The Arcsine Function

The sine function restricted to [−π/2,π/2] is one-to-one with range [−1,1] The inverse function is:

arcsin():[-1,1]→[−π/2,π/2]

By definition, arcsin(x)=θ means sin(θ)=x and −π/2≤θ≤π/2

Key values: arcsin(0)=0 arcsin(1/2)=π/6 arcsin(√(,2)/2)=π/4 arcsin(√(,3)/2)=π/3 arcsin(1)=π/2

Derivative of Arcsine

Using implicit differentiation on sin(arcsin(x))=x

d(arcsin(x))/d(x)=1/√(,1-x2), |x|<1

This formula is derived by noting that if y=arcsin(x) then cos(y)=√(,1−sin2(y))=√(,1−x2) (positive since y∈[−π/2,π/2]).

The Arccosine Function

The cosine function restricted to [0,π] is one-to-one with range [−1,1] The inverse is:

arccos():[−1,1]→[0,π]

The relationship arcsin(x)+arccos(x)=π/2 holds for all x∈[−1,1] reflecting the complementary angle identity sin(θ)=cos(π/2−θ)

Derivative of Arccosine

d(arccos(x))/d(x)=-1/√(,1-x2), |x|<1

The negative sign arises because arccosine is a decreasing function on its domain.

The Arctangent Function

The tangent function restricted to (−π/2,π/2) is one-to-one with range (−∞,∞) The inverse is:

arctan() :R→(−π/2,π/2)

Unlike arcsine and arccosine, arctangent is defined for all real numbers. It has horizontal asymptotes:

(lim_x→∞)(arctan(x))=π/2 and (lim_x→-∞)(arctan(x))=(-π)/2

Derivative of Arctangent

d(arctan(x))/d(x)=1/(1+x2)

This derivative, valid for all x leads to the important integral formula:

(∫_^)(1/(1+x2)*d(x))=arctan(x)+C

The Reciprocal Functions

Arcsecant

The secant function restricted appropriately gives:

arcsec(): (−∞,−1]∪[1,∞)→[0,π/2)∪(π/2,π]

The derivative is:

d(arcsec(x))/d(x)=1/|x|√(,x2−1))

Arccosecant

Similarly, arccosecant is defined on (−∞,−1]∪[1,∞) with range [−π/2,0)∪(0,π/2]

d(arccsc(x))/d(x)=−1/|x|√(,x2−1))

Arccotangent

The arccotangent function maps R to (0,π)

d(arccot(x))/d(x)=−1/(1+x2)

Important Identities

Complementary relationships connect inverse functions:

arcsin(x)+arccos(x)=π/2

arctan(x)+arccot(x)=π/2*(x>0)

arcsec(x)+arccsc(x)=π/2*(|x|≥1)

Negative arguments:

arcsin(−x)=−arcsin(x),arctan(−x)=−arctan(x)

arccos(−x)=π−arccos(x)

Integration Formulas

The derivatives of inverse trigonometric functions yield essential integration formulas:

(∫_^)(1/√(,1−x2)*d(x))=arcsin(x)+C

(∫_^)(1/(1+x2)*d(x))=arctan(x)+C

(∫_^)(1/(x√(,x2−1))*d(x))=arcsec(x)+C

More general forms with constants a>0

(∫_^)(1/√(,a2−x2)*d(x))=arcsin(x/a)+C

(∫_^)(1/(a2+x2)*d(x))=1/a*arctan(x/a)+C

Inverse Trig Substitution

Trigonometric substitutions transform integrals involving radicals into trigonometric integrals. The inverse functions then convert back to the original variable.

For √(,a2−x2) substitute x=a*sin(θ) so √(,a2−x2)=a*cos(θ)

For √(,a2+x2) substitute x=a*tan(θ) so √(,a2+x2)=a*sec(θ)

For √(,x2−a2) substitute x=a*sec(θ) so √(,x2−a2)=a*tan(θ)

Arctangent Addition Formula

For x*y<1

arctan(x)+arctan(y)=arctan((x+y)/(1−x*y))

This follows from the tangent addition formula. Special cases include:

arctan(1)+arctan(2)+arctan(3)=π

4*arctan(1/5)−arctan(1/239)=π/4 (Machin's formula)

Machin's formula and similar identities are used for computing digits of π

Series Expansions

The arctangent has a simple power series for |x|≤1

arctan(x)=x−(x3)/3+(x5)/5−(x7)/7+⋯=(∑_n=0^∞)(((−1)n*x(2*n+1))/(2*n+1))

At x=1 this gives the Leibniz formula:

π/4=1−1/3+1/5−1/7+⋯

Summary

The inverse trigonometric functions are defined by restricting domains to ensure one-to-one behavior. Arcsine maps [−1,1] to [−π/2,π/2] arccosine maps to [0,π] arctangent maps R to (−π/2,π/2)

Their derivatives are: d(arcsin(x))/d(x)=1/√(,1−x2) d(arccos(x))/d(x)=−1/√(,1−x2) and d(arctan(x))/d(x)=1/(1+x2) These formulas are essential for integration and appear throughout analysis.