This isn't too trigy :

Geometry Level 1

c o s 2 θ = 4 9 cos^2 \theta\ = \frac{4}{9} If the value of tan θ \theta can be given as: ± a b , \frac{\pm\sqrt{a}}{b}, where a a is a square free integer and b b is a positive integer. What is the value of: a + b ? a+b \ ?


The answer is 7.

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1 solution

Ryan Tamburrino
Feb 10, 2015

We see that sin 2 θ = 1 cos 2 θ = 1 4 9 = 5 9 \sin^2θ = 1 - \cos^2θ = 1 - \frac{4}{9} = \frac{5}{9} sin 2 θ cos 2 θ = tan 2 θ = 5 9 4 9 = 5 4 \frac{\sin^2θ}{\cos^2θ} = \tan^2θ = \frac{\frac{5}{9}}{\frac{4}{9}} = \frac{5}{4} tan θ = ± 5 2 a + b = 7 \tan θ = \pm \frac{\sqrt{5}}{2} \Rightarrow a+b = \boxed{7}

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