A geometry problem by Aly Ahmed

Geometry Level 3

2 tan ⁡ x + 3 tan ⁡ x + 6 tan ⁡ x = 1 \large 2^{\tan x} + 3^{\tan x} + 6^{\tan x} = 1

Find the smallest positive integer x x satisfying the equation above.

Clarification: Angles are measured in degrees.


The answer is 135.

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Aly Ahmed by Aly Ahmed , 34, Egypt

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2 solutions

Anurag Bhardewaj
Dec 9, 2016

Since 1/2 + 1/3 + 1/6 = 1 is a famous fraction so it was easy to figure out that value of tan X = -1 => x = arctan(-1) Since 0° < x <180° So x = 90°+ 45° = 135°

4 Helpful 0 Interesting 0 Brilliant 0 Confused

How do you know that tan ⁡ x = − 1 \tan x = -1 is the only possible solution?

Pi Han Goh - 4 years, 6 months ago
Tom Engelsman
Dec 8, 2016

Knowing that t a n ( 3 π 4 ) = − 1 tan(\frac{3\pi}{4}) = -1 , we obtain 2 t a n ( 3 π 4 ) + 3 t a n ( 3 π 4 ) + 6 t a n ( 3 π 4 ) = 1 2 + 1 3 + 1 6 = 1 2^{tan(\frac{3\pi}{4})} + 3^{tan(\frac{3\pi}{4})} + 6^{tan(\frac{3\pi}{4})} = \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = 1 .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

How would you know? Explain further that how you came to the value 135.

Sahil Silare - 4 years, 6 months ago

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