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Geometry Level 2

Given that tan ⁡ θ = 2 5 \tan \theta = \dfrac{2}{5} . Find the value of sin ⁡ 2 θ \sin 2 \theta exactly.

Bonus : Can you find the answer by using only the 3 basic functions ( sin ⁡ , cos ⁡ , tan ⁡ ) (\sin, \cos, \tan) .

3 8 \frac{3}{8} 20 29 \frac{20}{29} 5 29 \frac{5}{29} 19 29 \frac{19}{29}

Syed Hamza Khalid by Syed Hamza Khalid , 17, France

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1 solution

sin ⁡ 2 θ = 2 sin ⁡ θ cos ⁡ θ = 2 sin ⁡ θ cos ⁡ θ sec ⁡ 2 θ sec ⁡ 2 θ t = 2 tan ⁡ θ 1 + tan ⁡ 2 θ = 2 × 2 5 1 + ( 2 5 ) 2 = 20 29 \begin{aligned} \sin 2\theta & = 2 \sin \theta \cos \theta = \frac {2\sin \theta \cos \theta \sec^2 \theta}{\sec^2 \theta} t= \frac {2\tan \theta}{1+\tan^2 \theta} = \frac {2\times \frac 25}{1+\left(\frac 25\right)^2} = \boxed{\dfrac {20}{29}} \end{aligned}

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