A Nice Trig Formula
Grégoire Nicollier
University of Applied Sciences of Western Switzerland
November 26, 2013
$\tan50^\circ+\tan60^\circ+\tan70^\circ=\tan50^\circ\cdot\tan60^\circ\cdot\tan70^\circ=\tan80^\circ$
The first equality is well-known: iterate the identity
(1) | $\displaystyle\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$ |
to get
(2) | $\displaystyle\tan(\alpha+\beta+\gamma)=\frac{\tan\alpha+\tan\beta+\tan\gamma-\tan\alpha\tan\beta\tan\gamma}{1-\tan\alpha\tan\beta-\tan\alpha\tan\gamma-\tan\beta\tan\gamma}.$ |
Set $\alpha=50^\circ$, $\beta=60^\circ$, and $\gamma=70^\circ$ in (2), noticing that these angles sum up to $180^\circ$.
The second equality is less known: using (1) for $60^\circ\pm\delta$, $\tan60^\circ=\sqrt3$, and (2) for $\alpha=\beta=\gamma=\delta$, one gets $\tan (60^\circ-\delta)\tan (60^\circ+\delta)\tan\delta=\tan3\delta.$ Take then $\delta=10^\circ$.
Remark
The identity (2) shows that the true reason why $\arctan 1 + \arctan 2 + \arctan 3 = 180^\circ$ (see https://www.cut-the-knot.org/pythagoras/PaperFolding/arctan123.shtml) is that $1 + 2 + 3 = 1 \times 2 \times 3.$
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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