Ratio :)

Geometry Level 3

tan ⁡ θ = 1 3 \tan \theta = \dfrac{1}{3} ⟹ sin ⁡ θ = a b \implies \sin \theta = \dfrac{ \sqrt{a}}{b} or ⟹ sin ⁡ θ = − a b \implies \sin \theta = - \frac{ \sqrt{a}}{b} where a b \dfrac{ \sqrt{a}}{b} is in simplest form. Find a + b a+b

Details

Both a a and b b is a positive integers.


The answer is 20.

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Paul Ryan Longhas by Paul Ryan Longhas , 24, Philippines

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1 solution

Paul Ryan Longhas
Feb 20, 2015

tan ⁡ θ = 1 3 \tan \theta = \dfrac{1}{3} ⟹ sin ⁡ θ = ± 1 1 2 + 3 2 \implies \sin \theta = \pm \dfrac{1}{\sqrt{1^2 + 3^2}} ⟹ sin ⁡ θ = ± 10 10 \implies \sin \theta =\pm \dfrac{ \sqrt{10}}{10} Hence, a = 10 a = 10 and b = 10 b= 10 so, a + b = 10 + 10 = 20 a+b= 10+10= 20

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