Trigonometry easy

Geometry Level 3

Is the following statement always true?

If tan ⁡ x = 1 , tan ⁡ y = 1 2 , tan ⁡ z = 1 3 , then x = y + z . \text{If } \tan x = 1, \tan y = \dfrac12, \tan z= \dfrac13, \text{ then } x=y+z.

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พันดารา ธุวจิตต์ by พันดารา ธุวจิตต์ , 56, Thailand

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1 solution

Althought tan ⁡ ( y + z ) = tan ⁡ y + tan ⁡ z 1 − tan ⁡ y ⋅ tan ⁡ z = tan ⁡ x \tan(y+ z) = \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} = \tan x , it depends on which cuadrants the angles belongs to. Take y = tan ⁡ − 1 ( 1 2 ) , z = tan ⁡ − 1 ( 1 3 ) y = \tan^{-1} (\frac{1}{2}), \quad z = \tan^{-1} (\frac{1}{3}) belonging to the first cuadrant, then 0 < y < 45 º , 0 < z < 45 º ⇒ 0 < y + z < 90 º 0 < y < 45º, \space 0 < z < 45º \Rightarrow 0 < y + z < 90º and take x = 225 º x = 225º

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