Trigonometric Functions

Trigonometric Functions: Level 4 Challenges


k=1145[sin(2πk8)i cos(2πk14)]=1aicos(πb). \displaystyle \sum_{k=1}^{145} \left [ \sin \left ( \frac {2\pi k}{8} \right ) - i \ \cos \left ( \frac {2\pi k}{14} \right ) \right ] = \frac {1}{\sqrt{a}} - i \cos \left ( \frac {\pi}{b} \right ). Find the sum of the positive integers aa and b.b.

Note: i=1.i = \sqrt{-1}.

sec2(π9)+sec2(2π9)+sec2(4π9)=?\large \sec^{2} \left ( \dfrac{\pi}{9} \right ) + \sec^{2} \left (\dfrac{2 \pi}{9}\right) + \sec^{2} \left (\dfrac{4 \pi}{9}\right) = \, ?

Evaluate m=1tan1(8m16m432m2+5).\large \sum_{m=1}^{\infty} \tan^{-1} \left(\frac{8m}{16m^4-32m^2+5}\right).

Remember that the range of the inverse tangent function is (π2,π2) ( - \frac{ \pi}{2} , \frac{ \pi }{2}) .

If the value of sin24π24+cos24π24\sin^{24}\frac{\pi}{24} + \cos^{24}\frac{\pi}{24}

is expressed in simplest form as a+bcd, \dfrac{a + b\sqrt{c}}{d}, find the last three digits of a+b+c+d.a + b + c + d.

Note: Simplest form refers to the condition that a,b,c,da,b,c,d are positive integers with gcd(a,b,d)=1\gcd(a,b,d)=1 and cc is not divisible by the square of any prime.

If k=189cos6(k)=ab\displaystyle\sum_{k=1}^{89} \cos^{6}(k^{\circ}) = \frac{a}{b}, where aa and bb are positive coprime integers, then find a+b.a + b.


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