Question

How do you find the exact value of `sin85` using the sum and difference, double angle or half angle formulas?

Answer 1

`"The exact value of sin(85) cannot be found using sum and"` `"difference formulas or half-angle identities. The approximate"` `"decimal answer is 0.99619469."`

Explanation:

`cos(6x) = 32 cos^6(x) - 48 cos^4(x) + 18 cos^2(x) - 1`
`=> cos(6*85) = cos(510) = cos(150) = -sqrt(3)/2`
`= 32 cos^6(85) - 48 cos^4(85) + 18 cos^2(85) - 1`
`"name y = cos(85), then we have the sextic equation"`
`32 y^6 - 48 y^4 + 18 y^2 + sqrt(3)/2 - 1 = 0`
`"name z = y² then we have the cubic equation"`
`32 z³ - 48 z² + 18 z + sqrt(3)/2 - 1 = 0`
`"If one finds z analytically, one has z, y and sin(85) through"`
`sin^2(x)+cos^2(x) = 1.`
`"The problem with this cubic equation is that we have 3 real"`
`"roots so we need to use De Moivre to find the cuberoot because"`
`"the resulting quadratic equation has complex roots."`
`"Like this we are riding around in circles because De Moivre's"`
`"formula requires trig values."`
`"Anyway if you could obtain z in radicals in an analytical"`
`"way, you will make it to the math books and write history !"`