Geometry

Fundamental Trigonometric Identities

Fundamental Trigonometric Identities: Level 4 Challenges

         

Evaluate the sum

   logcos1(tan1)+logcos2(tan2)+logcos3(tan3)++logcos44(tan44)+logsin45(tan45)+logsin46(tan46)++logsin89(tan89).\begin{aligned} & \ \ \ \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{aligned}

Note: All angles are in degrees, and be aware that the base changes from cos \cos to sin \sin.

(7cosx+24sinx)(7sinx24cosx)\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.x.


Hint:

If xx and yy are acute angles such that

sinxsiny=12,cosxcosy=32, \frac {\sin x}{\sin y } = \frac {1}{2}, \quad \frac {\cos x}{\cos y } = \frac 3 2 ,

what is tan2(x+y)? \tan^2 (x+y)?

Define the function f(x)=2x1x2f(x)=\frac{2x}{1-x^2}. Find the number of distinct real solutions of the equation f(5)(x)=x.f^{(5)} (x) =x.

Details and assumptions

f(n)(x) f^{(n)} (x) denotes the function ff applied nn times. In particular, f(5)(x)=f(f(f(f(f(x))))) f^{(5)} (x) = f(f(f(f(f(x))))).

Find cos1˚cos2˚+cos2˚cos3˚++cos88˚cos89˚.\cos{1˚} \cos{2˚}+\cos{2˚} \cos{3˚}+ \cdots +\cos{88˚} \cos{89˚}.

Give your answer to two decimal places.

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