Geometry

Pythagorean Identities

Pythagorean Identities: Level 4 Challenges

         

{cosα=tanβcosβ=tanγcosγ=tanα \begin{cases} \cos \alpha = \tan \beta \\ \cos \beta = \tan \gamma \\ \cos \gamma = \tan \alpha \end{cases}

The acute angles α,β\alpha, \beta and γ\gamma satisfy the system of equations above. Find sinγ\sin \gamma.

Give your answer to 2 decimal places.

Find the value of E=tan2π40+tan23π40+tan25π40++tan219π40.E=\tan^2{\dfrac{\pi}{40}}+\tan^2{\dfrac{3\pi}{40}}+\tan^2{\dfrac{5\pi}{40}}+\cdots+\tan^2{\dfrac{19\pi}{40}}.

{sinθx=cosθycos4θx4+sin4θy4=97sin2θx3y+y3x \begin{cases} \dfrac{\sin\theta}{x} =\dfrac{\cos\theta}{y} \\ \dfrac{\cos^{4}\theta}{x^{4}}+\dfrac{\sin^{4}\theta}{y^{4}}=\dfrac{97\sin 2\theta}{x^{3}y+y^{3}x} \end{cases}

Let xx and yy be positive real numbers and θ\theta is an angle such that it is not a multiple of π2\frac{\pi}{2}. If x,yx,y and θ\theta satisfy the system of equations above, find xy+yx\dfrac{x}{y}+\dfrac{y}{x}.


Source: 2009 Harvard-MIT Mathematics Tournament

For all real values of θ\theta for which sinθ1,\lvert \sin \theta \rvert \neq 1, evaluate:

n=1sin2nθ.\large \sum_{n = 1}^{\infty} \sin^{2n}\theta .

For all real values of θ\theta for which sinθ1,\lvert \sin \theta \rvert \neq 1, evaluate:

n=1sin2nθ.\large \sum_{n = 1}^{\infty} \sin^{2n}\theta .

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