JEE Determinants 3

Geometry Level 5

3x2x2+xcosθ+cos2θx2+xsinθ+sin2θx2+xcosθ+cos2θ3cos2θ1+sin2θ2x2+xsinθ+sin2θ1+sin2θ23sin2θ=0\left| \begin{matrix} 3{ x }^{ 2 } & x^{ 2 }+x \cos\theta +{ cos }^{ 2 }\theta & x^{ 2 }+x \sin\theta +{ \sin }^{ 2 }\theta \\ x^{ 2 }+x \cos\theta +{ \cos }^{ 2 }\theta & 3 \cos^{ 2 }\theta & 1+\frac { \sin2\theta }{ 2 } \\ x^{ 2 }+x \sin\theta+{ \sin }^{ 2 }\theta & 1+\frac { \sin2\theta }{ 2 } & 3{ \sin }^{ 2 }\theta \end{matrix} \right| =0 are

Given the roots of the above determinant are f(θ)f(\theta) and g(θ)g(\theta). Evaluate (f(π29))2+(g(π29))2(f(\frac{\pi}{29}))^{2}+(g(\frac{\pi}{29}))^{2}

You may use calculator for the calculations.

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