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Pythagorean Identities

Trigonometric identities bring new life to the Pythagorean theorem by re-envisioning the legs of a right triangle as sine and cosine.

Pythagorean Identities: Level 4 Challenges


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

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

Find the value of \[E=\tan^2{\dfrac{\pi}{40}}+\tan^2{\dfrac{3\pi}{40}}+\tan^2{\dfrac{5\pi}{40}}+\cdots+\tan^2{\dfrac{19\pi}{40}}.\]

\[ \begin{cases} \cos \alpha = \tan \beta \\ \cos \beta = \tan \gamma \\ \cos \gamma = \tan \alpha \end{cases} \]

The acute angles \(\alpha, \beta\) and \(\gamma\) satisfy the system of equations above. Find \(\sin \gamma\).

Give your answer to 2 decimal places.

\[ \begin{cases} \dfrac{\sin\theta}{x} =\dfrac{\cos\theta}{y} \\ \dfrac{\cos^{4}\theta}{x^{4}}+\dfrac{\sin^{4}\theta}{y^{4}}=\dfrac{97\sin 2\theta}{x^{3}y+y^{3}x} \end{cases} \]

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

Source: 2009 Harvard-MIT Mathematics Tournament

If \(x\) and \(y\) are acute angles such that

\[ \frac {\sin x}{\sin y } = \frac {1}{2}, \quad \frac {\cos x}{\cos y } = \frac 3 2 , \]

what is \( \tan^2 (x+y) \)?


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