Geometry

Pythagorean Identities

Pythagorean Identities: Level 3 Challenges

         

Given that cos35°=α\cos{35°}=\alpha, express sin2015\sin{2015^\circ} in terms of α\alpha.


This problem is part of the set 2015 Countdown Problems.

1sin2θ1cos2θ1tan2θ1cot2θ1sec2θ1csc2θ=3\dfrac1{\sin^2{\theta}}-\dfrac1{\cos^2{\theta}}-\dfrac1{\tan^2{\theta}}-\dfrac1{\cot^2{\theta}}-\dfrac1{\sec^2{\theta}}-\dfrac1{\csc^2{\theta}}=-3

Find the number of solutions of θ\theta in the interval (0,2π)(0,2\pi) that satisfy the equation above.

4sin2(x)+tan2(x)+cot2(x)+csc2(x)=6 \large 4\sin^2 (x) + \tan^2(x) + \cot^2(x) + \csc^2(x) = 6

Find the number of solutions of xx in the interval [0,2π][0,2\pi] that satisfy the equation above.

For real number θ\theta satisfying 1+2sin2θ=75cos3θ 1 + 2 \sin^2{\theta} = 75 \, \cos^3{\theta} , what is the value of 3+4tan4θ3 + 4 \tan^4{\theta} ?

As xx ranges over all real values, what is the range of

A=sin4x+cos2x? A=\sin^4x+\cos^2x?

If you're looking to skyrocket your preparation for JEE-2015, then go for solving this set of questions.
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