A geometry problem by Vivek Vijayan

Geometry Level 3

If 2 − cos ⁡ 2 θ = 3 sin ⁡ θ cos ⁡ θ 2-\cos^{2} \theta = 3\sin \theta \cos \theta , sin ⁡ θ ≠ cos ⁡ θ \sin \theta \not= \cos \theta then tan ⁡ θ \tan \theta is


The answer is 0.5.

Vivek Vijayan by Vivek Vijayan , 34, India

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1 solution

Ronak Agarwal
Jul 14, 2014

D i v i d i n g b o t h s i d e s b y c o s 2 θ w e g e t 2 s e c 2 θ − 1 = 3 t a n θ ⇒ 2 t a n 2 θ − 3 t a n θ + 1 = 0 ⇒ ( 2 t a n θ − 1 ) ( t a n θ − 1 ) = 0 S i n c e t a n θ ≠ 1 ⇒ t a n θ = 1 / 2 Dividing\quad both\quad sides\quad by\quad { cos }^{ 2 }\theta \quad we\quad get\quad \\ 2{ sec }^{ 2 }\theta -1=3tan\theta \quad \Rightarrow 2{ tan }^{ 2 }\theta -3tan\theta +1=0\\ \Rightarrow (2tan\theta -1)(tan\theta -1)=0\quad Since\quad tan\theta \neq 1\\ \Rightarrow tan\theta =1/2

2 Helpful 0 Interesting 0 Brilliant 0 Confused

I rewrote the expression as 2 s i n 2 θ + c o s 2 θ = 3 s i n θ . c o s θ 2 sin^{2}\theta + cos^{2}\theta= 3sin\theta.cos\theta ..later divided it by cos^2 theta and then wrote it as a quadratic in t a n θ tan\theta and got it...because I didnt think of dividing it in the first itself by cos^2 :P

Jayakumar Krishnan - 6 years, 9 months ago

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Same way Jaya! :D

Krishna Ar - 6 years, 9 months ago

Great problem @Vivek Vijayan ...something like this could come up in my SA1...wish...it does :P

Jayakumar Krishnan - 6 years, 9 months ago

same way as I did!

Kartik Sharma - 6 years, 9 months ago

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