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Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?
Infinite Continued Fractions
In this article we are going to look at infinite continued fractions - continued fractions that do not terminate.
Complex Sine
Solve the equation sin z = 2 for complex z. You only need the formula you are given for sin z in terms of the exponential function, and to solve a quadratic equation and use the logarithmic function.
Age 16 to 18
Challenge Level
The hyperbolic trig functions $\cosh $ and $\sinh $ are defined by $$\eqalign { \cosh x &= {1\over 2}(e^x + e^{-x}) \cr \sinh x &= {1\over 2}(e^x - e^{-x}).}$$ Using the definitions sketch the graphs of $\cosh x$ and $\sinh x$ on one diagram and prove the hyperbolic trig identities $$\eqalign { \cosh^2 x - \sinh^2 x &=1 \cr \sinh 2x &= 2\sinh x \cosh x \cr \sinh (n+1)x &= \sinh nx \cosh x + \cosh nx \sinh x.}$$
Notice the strong resemblance of these formulae to standard trigonometrical identities. Using this similarity as a guide, investigate the properties of a 'hyperbolic tangent' function $tanh(x)$ defined by
$$\tanh(x)=\frac{\sinh(x)}{\cosh(x)}$$
NOTES AND BACKGROUND
Notice that the identities for hyperbolic functions that you have proved are very similar to the ordinary trigonometric identities. In fact there is a complete hyperbolic geometry with similar results to the trigonometric results in Euclidean geometry. We compare absolute values in the corresponding result for $\sin nx$ which is $|\sin nx|\leq n|\sin x|$ . This formula needs the absolute values because the function is periodic and takes negative values for some multiples of the angle. Notice that the inequality in $|\sin nx|\leq n|\sin x|$ goes the other way to the corresponding hyperbolic result. This is because $\cos x \leq 1$ for all $x$ whereas $\cosh x\geq 1$.
NRICH is part of the family of activities in the Millennium Mathematics Project.