One Trigonometric Formula and Its Consequences
The formula below that holds for any angles $\alpha,$ $\beta,$ $\gamma$ has a two step proof and great many consequences:
(*)
$\displaystyle\sin\alpha +\sin \beta +\sin \gamma -\sin (\alpha + \beta + \gamma )=4\sin\frac{\alpha +\beta}{2}\sin\frac{\beta +\gamma}{2}\sin\frac{\gamma +\alpha}{2}.$
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Copyright © 1996-2018 Alexander Bogomolny
From the Addition formulas for sine and cosine one can easily get important formulas for the sum and difference of two sines or of two cosines:
(1)
$\displaystyle\sin\alpha +\sin \beta = 2\sin\frac{\alpha +\beta}{2}\cos\frac{\alpha -\beta}{2},$
(2)
$\displaystyle\sin\alpha -\sin \beta = 2\sin\frac{\alpha -\beta}{2}\cos\frac{\alpha +\beta}{2},$
(3)
$\displaystyle\cos\alpha +\cos \beta = 2\cos\frac{\alpha +\beta}{2}\cos\frac{\alpha -\beta}{2},$
(4)
$\displaystyle\cos\alpha -\cos \beta = 2\sin\frac{\alpha +\beta}{2}\sin\frac{\beta-\alpha}{2},$
For example, since
$\sin(a \pm b) =\sin (a) \cos (b) \pm \cos (a)\sin (b)$
then
$\sin(a + b) +\sin (a - b) = 2\sin (a) \cos (b).$
Now (1) follows by setting $\alpha = a + b$ and $\beta = a - b$ and solving the system for $a$ and $b:$
$\displaystyle\begin{align} a &= (\alpha + \beta ) / 2,\\ b &= (\alpha - \beta ) / 2. \end{align}$
To prove (*),
(*)
$\displaystyle\sin\alpha +\sin \beta +\sin \gamma -\sin (\alpha + \beta + \gamma )=4\sin\frac{\alpha +\beta}{2}\sin\frac{\beta +\gamma}{2}\sin\frac{\gamma +\alpha}{2}.$
we apply (1) and (2) to the left side and subsequently (4) to the result:
$\displaystyle\begin{align} \sin\alpha &+\sin \beta +\sin \gamma -\sin (\alpha + \beta + \gamma )\\ &=2\sin\frac{\alpha + \beta}{2} \cos\frac{\alpha - \beta}{2} - 2\sin\frac{\alpha + \beta}{2} \cos\frac{\gamma + (\alpha + \beta )}{2}\\ &=2\sin\frac{\alpha + \beta}{2} \left[\cos\frac{\alpha - \beta}{2} - \cos\frac{\gamma + (\alpha + \beta )}{2}\right]\\ &=4\sin\frac{\alpha + \beta}{2}\sin\frac{\beta + \gamma}{2}\sin\frac{\gamma + \alpha}{2}, \end{align}$
as required.
When $\alpha,$ $\beta,$ $\gamma$ are angles of a triangle, i.e., when, in particular, $\alpha + \beta + \gamma = \pi,$ we may observe that, for example,
$\displaystyle\frac{\alpha + \beta}{2} = \frac{\pi}{2} - \frac{\gamma}{2},$
which allows to reduces (*) to
(**)
$\displaystyle\sin\alpha +\sin \beta +\sin \gamma = 4 \cos\frac{\alpha}{2} \cos\frac{\beta}{2} \cos\frac{\gamma}{2},$
because $\displaystyle\sin (\frac{\pi}{2} - x) = \cos (x).$
The latter identity (**) has further consequences. Right now, I'll give a second, more direct proof of (**) [Stanford Problem Book, #47.4]. It's not easier than the above and I do not know if it admits a simpler proof than the more general (*).
Let $\alpha = 2u,$ $\beta = 2v,$ $\gamma = \pi - 2u - 2v.$ Then, using the formula for the double argument $\sin(2x) = 2\sin (x)\cos (x),$ (**) is transformed into
$\begin{align} \sin(2u) +\sin (2v) &= 2[2\cos (u)\cos (v) - \cos (u + v)]\sin(u + v)\\ &= 2[\cos (u)\cos (v) +\sin (u)\sin(v)]\sin(u + v)\\ &= 2\cos (u - v)\sin (u + v) \end{align}$
which is actually (1) and hence true.
Reference
- G. Polya, J. Kilpatrick, The Stanford Mathematics Problem Book, Dover, 2009
Trigonometry
- What Is Trigonometry?
- Addition and Subtraction Formulas for Sine and Cosine
- The Law of Cosines (Cosine Rule)
- Cosine of 36 degrees
- Tangent of 22.5o - Proof Wthout Words
- Sine and Cosine of 15 Degrees Angle
- Sine, Cosine, and Ptolemy's Theorem
- arctan(1) + arctan(2) + arctan(3) = π
- Trigonometry by Watching
- arctan(1/2) + arctan(1/3) = arctan(1)
- Morley's Miracle
- Napoleon's Theorem
- A Trigonometric Solution to a Difficult Sangaku Problem
- Trigonometric Form of Complex Numbers
- Derivatives of Sine and Cosine
- ΔABC is right iff sin²A + sin²B + sin²C = 2
- Advanced Identities
- Hunting Right Angles
- Point on Bisector in Right Angle
- Trigonometric Identities with Arctangents
- The Concurrency of the Altitudes in a Triangle - Trigonometric Proof
- Butterfly Trigonometry
- Binet's Formula with Cosines
- Another Face and Proof of a Trigonometric Identity
- cos/sin inequality
- On the Intersection of kx and |sin(x)|
- Cevians And Semicircles
- Double and Half Angle Formulas
- A Nice Trig Formula
- Another Golden Ratio in Semicircle
- Leo Giugiuc's Trigonometric Lemma
- Another Property of Points on Incircle
- Much from Little
- The Law of Cosines and the Law of Sines Are Equivalent
- Wonderful Trigonometry In Equilateral Triangle
- A Trigonometric Observation in Right Triangle
- A Quick Proof of cos(pi/7)cos(2.pi/7)cos(3.pi/7)=1/8
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