lmao
Lv 6
lmao asked in Science & MathematicsMathematics · 9 years ago

Is this an identity?...........?

cos(3x)= 4(cos(x))^3-3cos(x)

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  • 9 years ago
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    Yes, it is. One way to see it is to apply the angle addition formulas

    cos(a+b) = cos(a) cos(b) - sin(a) sin(b)

    sin(a+b) = sin(a) cos(b) + cos(a) sin(b)

    a few times:

    cos(3x) = cos(x + 2x)

    = cos(x) cos(2x) - sin(x) sin(2x)

    = cos(x) cos(x+x) - sin(x) sin(x+x)

    = cos(x) (cos(x) cos(x) - sin(x) sin(x)) - sin(x) (sin(x) cos(x) + cos(x) sin(x))

    = (cos(x))^3 - cos(x) (sin(x))^2 - 2 (sin(x))^2 cos(x)

    ... then use the fact that (sin(x))^2 = 1 - (cos(x))^2 ...

    = (cos(x))^3 - cos(x) (1 - (cos(x))^2) - 2 (1 - (cos(x))^2) cos(x)

    = (cos(x))^3 - cos(x) + (cos(x))^3 - 2 cos(x) + 2 (cos(x))^3

    = 4 (cos(x))^3 - 3 cos(x)

    Like any trig identity there are other derivations also but that is one of them.

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