Definition:Sine/Real Function

Definition

The real sine function $\sin: \R \to \R$ is defined as:

\(\ds \forall x \in \R: \, \)

\(\ds \sin x\)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}\)

\(\ds \)

\(=\)

\(\ds x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots\)

Graph of Sine Function

Arch of Sine Function

Each section of the sine function between adjacent zeroes is called an arch of the sine function

Also see

• Sine Function is Absolutely Convergent

• Definition:Real Cosine Function

• Results about the sine function can be found here.

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables: $20.21$
• 1969: J.C. Anderson, D.M. Hum, B.G. Neal and J.H. Whitelaw: Data and Formulae for Engineering Students (2nd ed.) ... (previous) ... (next): $4.$ Mathematics: $4.2$ Series
• 1973: G. Stephenson: Mathematical Methods for Science Students (2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.3$ Functions of a Real Variable
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 16.2$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $24$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $24$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): expansion
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): trigonometric function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): expansion: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): trigonometric function