Alphabeta MathFurther Trigonometric Identities and Inverse Functions
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1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- Equivalent Forms of Completeness
- Foundations of the Real Numbers for Analysis
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Principal inverse sine and inverse cosine
Theorem
For -1<y<1, (arcsin y)ᵖʳⁱᵐᵉ=1/√1-y² and (arccos y)ᵖʳⁱᵐᵉ=-1/√1-y²
Lemma
Tangent is a continuous strictly increasing bijection from (-π/2,π/2) onto ℝ
Definition
The principal inverse tangent arctan:ℝ→(-π/2,π/2)
Theorem
Principal arctangent: derivative, integral, power series, and the Gregory–Leibniz series
Theorem
Standard Maclaurin expansions
Sources
- NIST Digital Library of Mathematical Functions, Inverse Trigonometric Functions
- NIST Digital Library of Mathematical Functions, §4.23 Inverse Trigonometric Functions
- NIST Digital Library of Mathematical Functions, §4.23–4.24
- J. Lebl, Basic Analysis I, §4.4 Inverse function theorem
- W. F. Trench, Introduction to Real Analysis, §4.5, pp. 265–267
- MIT OpenCourseWare 18.100C Real Analysis, Lecture 23