complexideas

Algebra Level 2

If x + 1 x = 58 i x+\dfrac 1x = \sqrt{58}i , then it can be proven that sin ⁡ ( x 2 ) + sin ⁡ ( 1 x 2 ) + cos ⁡ ( m ) = 0 \sin (x^2) + \sin \left(\dfrac 1{x^2}\right) + \cos (m) = 0 , where m m is a positive integer. Find the value of cos ⁡ ( m + 2 ) \cos(m+2) .

Notation: i = − 1 i = \sqrt{-1} denotes the imaginary unit .


The answer is 1.

Aziz Alasha by Aziz Alasha , 59, Sudan

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1 solution

Aziz Alasha
Jul 19, 2017

X + 1/x = (√58)i , x² + 1/x² = -58-2 = -60 = M , M/2 = -30 , x² - 1/x² = 60² - 4 =3596 = N , N/2 =1798, Sin(x²)+Sin(1/x²) = 2SinM/2CosN/2 = 2Sin-30Cos1798 = -cos1798 , Sin(x²)+Sin(1/x²)+cos1798 = 0 , m = 1798 , m+2 = 1800 = 5(360) , Cos(m+2) = 1.

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