I was working through a booklet of Olympiad-style problems when I came across a method which used the substitution $x = \cos \alpha$ to solve $x = \sqrt{2 + \sqrt{2-\sqrt{2+x}}}$. The solution works out nicely using the half angle formula. Are there any other good examples of such equations, where a trigonometric substitution and an identity can reduce a problem like this so effectively?
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1$\begingroup$ you can take a look at AOPS (art of problemsolving) $\endgroup$– Dr. Sonnhard GraubnerJul 6, 2017 at 9:22
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1$\begingroup$ A neat way to maximize $f(x) = \alpha x(1-x)$ (logistic mapping) on the interval $[0,1]$ without calculus is setting $x = \sin^2 \theta$ so we get $\alpha\sin^2\theta\cos^2\theta = \frac{\alpha}{4} \sin^2 2\theta$ $\endgroup$– MathematicsStudent1122Jul 6, 2017 at 9:30
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$\begingroup$ See also: math.stackexchange.com/questions/429883/roots-of-8x3-4x2-4x1 and math.stackexchange.com/questions/550052/… math.stackexchange.com/questions/2203364/… math.stackexchange.com/questions/2157643/… $\endgroup$– lab bhattacharjeeJul 6, 2017 at 9:42
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1 Answer
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The substitution
$$t=a\cos^2 \theta+b \sin^2 \theta$$
Simplifies the function $\displaystyle \sqrt{\frac{t - a}{b - t}}$ tremendously to $\tan \theta$.
You can see application of this substitution here
answered Jul 6, 2017 at 9:37
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$\begingroup$ This may look artificial until you know the derivation : math.stackexchange.com/questions/2304904/… $\endgroup$ Jul 6, 2017 at 9:40
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$\begingroup$ @labbhattacharjee I don't understand the meaning of artificial. Can you please elaborate? $\endgroup$ Jul 6, 2017 at 9:43
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$\begingroup$ How on earth, you know it and how much it is susceptible to generalization? $\endgroup$ Jul 6, 2017 at 9:44
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$\begingroup$ @labbhattacharjee Actually, it was taught to us in "Indefinite Integration $\rightarrow$ Integration by substitution". I have memorized it as it is and never knew about it's 'derivation'. $\endgroup$ Jul 6, 2017 at 9:48
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$\begingroup$ Exactly, therein lies the problem! "How/why" is missing in many places in our curriculum. For example, many know how to check for divisibility by $3$ but how many know why the rule works? math.stackexchange.com/questions/328562/… $\endgroup$ Jul 6, 2017 at 9:52