Sine of Sum
Theorem
- $\map \sin {a + b} = \sin a \cos b + \cos a \sin b$
where $\sin$ denotes the sine and $\cos$ denotes the cosine.
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Corollary
- $\map \sin {a - b} = \sin a \cos b - \cos a \sin b$
Proof 1
\(\ds \map \cos {a + b} + i \, \map \sin {a + b}\) | \(=\) | \(\ds e^{i \paren {a + b} }\) | Euler's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{i a} e^{i b}\) | Exponential of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\cos a + i \sin a} \paren {\cos b + i \sin b}\) | Euler's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\cos a \cos b - \sin a \sin b} + i \paren {\sin a \cos b + \cos a \sin b}\) | Complex Numbers form Field |
By equating the imaginary parts, the result follows.
$\blacksquare$
Proof 2
Recall the analytic definitions of sine and cosine:
- $\ds \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}$
- $\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}$
Let:
\(\ds \map g a\) | \(=\) | \(\ds \map \sin {a + b} - \sin a \cos b - \cos a \sin b\) | ||||||||||||
\(\ds \map h a\) | \(=\) | \(\ds \map \cos {a + b} - \cos a \cos b + \sin a \sin b\) |
Let us differentiate these with respect to $a$, keeping $b$ constant.
Then from Derivative of Sine Function and Derivative of Cosine Function, we have:
\(\ds \map {g'} a\) | \(=\) | \(\ds \map \cos {a + b} - \cos a \cos b + \sin a \sin b = \map h a\) | ||||||||||||
\(\ds \map {h'} a\) | \(=\) | \(\ds -\map \sin {a + b} + \sin a \cos b + \cos a \sin b = -\map g a\) |
Hence:
\(\ds \map {D_a} {\paren {\map g a}^2 + \paren {\map h a}^2}\) | \(=\) | \(\ds 2 \map g a \map {g'} a + 2 \map h a \map {h'} a\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
Thus from Derivative of Constant:
- $\forall a \in \R: \map g a^2 + \map h a^2 = c$
In particular, it is true for $a = 0$, and so:
- $\map g 0^2 + \map h 0^2 = 0$
So:
- $\map g a^2 + \map h a^2 = 0$
But from Square of Real Number is Non-Negative:
- $\map g a^2 \ge 0$ and $\map h a^2 \ge 0$
So it follows that:
- $\map g a = 0$
and:
- $\map h a = 0$
Hence the result.
$\blacksquare$
Proof 3
\(\ds \sin a \cos b + \cos a \sin b\) | \(=\) | \(\ds \paren {\frac {e^{i a} - e^{-i a} }{2 i} } \cos b + \cos a \paren {\frac {e^{i b} - e^{-i b} }{2 i} }\) | Euler's Sine Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {e^{i a} - e^{-i a} } {2 i} } \paren {\frac {e^{i b} + e^{-i b} } 2} + \paren {\frac {e^{i a} + e^{-i a} } 2} \paren {\frac {e^{i b} - e^{-i b} } {2 i} }\) | Euler's Cosine Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{i a} e^{i b} + e^{i a} e^{-i b} - e^{-i a} e^{i b} - e^{-i a} e^{-i b} } {4 i}\) | expanding | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac {e^{i a} e^{i b} - e^{i a} e^{-i b} + e^{-i a} e^{i b} - e^{-i a} e^{-i b} } {4 i}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{i a} e^{i b} - e^{-i a} e^{-i b} } {2 i}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{i \paren {a + b} } - e^{-i \paren {a + b} } } {2 i}\) | Exponential of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \sin {a + b}\) | Euler's Sine Identity |
$\blacksquare$
Proof 4
\(\ds \sin \left({a + b}\right)\) | \(=\) | \(\ds \cos \left({\frac \pi 2 - \left({a + b}\right)}\right)\) | Cosine of Complement equals Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \cos \left({\left({\frac \pi 2 - a}\right) - b}\right)\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cos \left({\frac \pi 2 - a}\right) \cos b + \sin \left({\frac \pi 2 - a}\right) \sin b\) | Cosine of Difference | |||||||||||
\(\ds \) | \(=\) | \(\ds \sin a \cos b + \sin \left({\frac \pi 2 - a}\right) \sin b\) | Cosine of Complement equals Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \sin a \cos b + \cos a \sin b\) | Sine of Complement equals Cosine |
$\blacksquare$
Proof 5
\(\text {(1)}: \quad\) | \(\ds 2 \sin a \cos b\) | \(=\) | \(\ds \sin \paren {a + b} + \sin \paren {a - b}\) | Werner Formula for Sine by Cosine: Proof 2 | ||||||||||
\(\text {(2)}: \quad\) | \(\ds 2 \cos a \sin b\) | \(=\) | \(\ds \sin \paren {a + b} - \sin \paren {a - b}\) | Werner Formula for Cosine by Sine: Proof 2 | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \sin \paren {a + b}\) | \(=\) | \(\ds 2 \sin a \cos b + 2 \cos a \sin b\) | $(1) + (2)$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sin \paren {a + b}\) | \(=\) | \(\ds \sin a \cos b + \cos a \sin b\) |
$\blacksquare$
Proof 6
We begin by enclosing a right-angled triangle $BEF$ with hypotenuse $BF$ of length $1$, inside rectangle $ABCD$.
Let $\angle EBF = a$ and $\angle ABE = b$.
Therefore:
\(\ds BF\) | \(=\) | \(\ds 1\) | Given | |||||||||||
\(\ds BE\) | \(=\) | \(\ds \cos a\) | Definition of Cosine of Angle | |||||||||||
\(\ds EF\) | \(=\) | \(\ds \sin a\) | Definition of Sine of Angle | |||||||||||
\(\ds AB\) | \(=\) | \(\ds \cos a \cos b\) | ||||||||||||
\(\ds AE\) | \(=\) | \(\ds \cos a \sin b\) | ||||||||||||
\(\ds ED\) | \(=\) | \(\ds \sin a \cos b\) | ||||||||||||
\(\ds DF\) | \(=\) | \(\ds \sin a \sin b\) | ||||||||||||
\(\ds \map \sin {a + b }\) | \(=\) | \(\ds BC\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds AE + ED\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cos a \sin b + \sin a \cos b\) |
$\blacksquare$
Proof 7
Let two triangles $\triangle ABC$ and $\triangle ADC$ be inscribed in a circle on opposite sides of diameter $AC$.
By Thales' Theorem, they are both right triangles and $\angle ADC$ and $\angle ABC$ are right angles.
Let the diameter $AC = 1$.
Let $\angle DAC = \alpha$ and $\angle CAB = \beta$.
From the construction above, we have the following:
- $\cos \alpha = AD$
- $\cos \beta = AB$
- $\sin \alpha = DC$
- $\sin \beta = BC$
- $DB \mathop = 2 r \map \sin {\alpha + \beta}$
Since $2r \mathop = 1$:
- $DB \mathop = \map \sin {\alpha + \beta}$
By Quadrilateral is Cyclic iff Opposite Angles sum to Two Right Angles:
- $\Box ABCD$ is a cyclic quadrilateral.
- $DB \times AC = AB \times CD + BC \times AD$
Substituting:
- $\map \sin {\alpha + \beta} \times 1 = \cos \beta \sin \alpha + \sin \beta \cos \alpha$
- $\map \sin {\alpha + \beta} = \sin \alpha \cos \beta + \sin \beta \cos \alpha$
By Equivalence of Definitions of Sine of Angle, the definition of sine from the circle, from the triangle and as a real function are equivalent.
It follows that all real numbers $x$ and $y$ correspond to values of $\alpha$ and $\beta$ for which the proof above applies, with one exception.
The exception occurs when both $\alpha$ and $\beta$ are equal to $\dfrac {\pi} 2$.
But then the result is simply:
- $ \sin {\pi} = \sin \dfrac {\pi} 2 \cos \dfrac {\pi} 2 + \sin \dfrac {\pi} 2 \cos \dfrac {\pi} 2$
- $ 0 = 0 \cdot 1 + 1 \cdot 0$
The result follows.
$\blacksquare$
Historical Note
The Sine of Sum formula and its corollary were proved by François Viète in about $1579$.
Also see
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(12)$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.34$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $4$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): addition formula: 1.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): addition formulae
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): addition formulae
- 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $5$: Eternal Triangles: Ptolemy
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): compound angle formulae (in trigonometry)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Addition formulae
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Addition formulae