# A Formula for sin(3x)

The prupose of this page is to prove the following formula:

$\sin 3x =4\sin x\sin(60^{\circ}-x)\sin(60^{\circ}+x).$

We first remind of another useful trigonometric identity:

$\displaystyle\sin\alpha + \sin\beta +\sin\gamma -\sin(\alpha +\beta +\gamma)=4\sin\frac{\alpha +\beta}{2}\sin\frac{\beta +\gamma}{2}\sin\frac{\gamma +\alpha}{2}.$

Taking here $\alpha =x,$ $\beta =x+120^{\circ},$ $\gamma =x-120^{\circ}$ reduces the formula to

$\sin x + \sin (x+120^{\circ}) +\sin(x-120^{\circ}) -\sin 3x=4\sin x \sin (x+60^{\circ})\sin (x-60^{\circ}),$

implying that in order to establish the initial identity, we need to show that

$\sin x + \sin (x+120^{\circ}) +\sin(x-120^{\circ})=0.$

The easiest way to do that is to recourse to complex numbers and Euler's formula $e^{i\theta}=\cos\theta +\sin\theta.$ In complex numbers, the three cube roots of unity add up to $0:$

$1 + e^{i\frac{\pi}{3}}+ + e^{-i\frac{\pi}{3}} = 0.$

This multiplied by any factor still remains $0:$

$e^{ix} + e^{i(x+\frac{\pi}{3})}+ + e^{i(x-\frac{\pi}{3})} = 0.$

But, if a complex number is $0$, both its real and imaginary parts also vanish. For the imaginary part, we have exactly what's needed:

$\displaystyle\sin x + \sin (x+\frac{\pi}{3}) +\sin(x-\frac{\pi}{3})=0.$

### Trigonometry

- What Is Trigonometry?
- Addition and Subtraction Formulas for Sine and Cosine
- The Law of Cosines (Cosine Rule)
- Cosine of 36 degrees
- Tangent of 22.5
^{o}- 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

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