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Fundamental Trigonometric Identities
These are your basic building blocks for solving trigonometric equations and understanding how the pieces fit together. Using these identities can make sense of even the scariest looking trig.
Evaluate the sum
\[\begin{align} & \ \ \ \log_{\cos1}(\tan{1}) \\ &+\log_{\cos{2}}(\tan{2}) \\ &+ \log_{\cos{3}}(\tan{3}) \\& +\ldots \\ &+ \log_{\cos{44}} (\tan{44}) \\ &+ \log_{\sin{45}}(\tan{45}) \\ &+ \log_{\sin46}(\tan{46}) \\ &+\ldots \\ &+ \log_{\sin89}(\tan{89}). \end{align}\]
Note: All angles are in degrees, and be aware that the base changes from \( \cos \) to \( \sin\).
by
Trevor Arashiro
\[\large (7\cos x+24\sin x)(7\sin x-24\cos x) \]
Find the maximum value of this expression over all real values \(x.\)
Hint:
by
avn bha
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 \( \tan^2 (x+y) \)?
by
Guru Prasaadh
Define the function \(f(x)=\frac{2x}{1-x^2}\). Find the number of distinct real solutions of the equation \(f^{(5)} (x) =x.\)
Details and assumptions
\( f^{(n)} (x) \) denotes the function \(f\) applied \(n\) times. In particular, \( f^{(5)} (x) = f(f(f(f(f(x)))))\).
by
Calvin Lin
Find \[\cos{1˚} \cos{2˚}+\cos{2˚} \cos{3˚}+ \cdots +\cos{88˚} \cos{89˚}.\]
Give your answer to two decimal places.
by
Jonathan Schirmer