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Math Help - Deriving an equation using Trig formulas

  1. #1
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    Question Deriving an equation using Trig formulas

    I'm fairly confident in using the Pythagorean, Sum, Difference, Double Angle and half Angle formulas. The problem is that I struggle with the mental gymnastics to demonstrate sin(3x) = something horrendous, in truth I spend far too much time pursuing blind alleys that look right for far too long.

    I can get as far as using the Sum rule to break the Sin(3x) into say

    Sin(2x + x) = sin(2x)cos(x)+cos(2x)sin(x) = (2sin(x)cos(x))cos(x)+(cos^2(x)-sin^2(x))sin(x)

    But getting it into the final form always seems a step too far.

    Can anyone offer some pointers as to how you would go about solving such problems (the only example I have is an assignment question so it would be dishonest of me to post a variation on it for help).
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  2. #2
    Super Member Quacky's Avatar
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    Quote Originally Posted by AliceFisher View Post
    I'm fairly confident in using the Pythagorean, Sum, Difference, Double Angle and half Angle formulas. The problem is that I struggle with the mental gymnastics to demonstrate sin(3x) = something horrendous, in truth I spend far too much time pursuing blind alleys that look right for far too long.

    I can get as far as using the Sum rule to break the Sin(3x) into say

    Sin(2x + x) = sin(2x)cos(x)+cos(2x)sin(x) = (2sin(x)cos(x))cos(x)+(cos^2(x)-sin^2(x))sin(x)

    But getting it into the final form always seems a step too far.

    Can anyone offer some pointers as to how you would go about solving such problems (the only example I have is an assignment question so it would be dishonest of me to post a variation on it for help).
    You're almost done! Remember that you can rewrite Cos^2(x)-Sin^2(x) as (1-2Sin^2(x)) or (2Cos^2(x)-1) - it will help in some situations (I haven't used it here, although I thought it was worth mentioning).

    (2Sin(x)Cos(x))Cos(x)+(Cos^2(x)-Sin^2(x))Sin(x) .

    So multiply through:
    2Sin(x)Cos^2(x)+Cos^2(x)Sin(x)-Sin^3(x)

    Collect like terms:
    =3Sin(x)Cos^2(x)-Sin^3(x)

    =3Sin(x)(1-Sin^2(x))-Sin^3(x) (see below for explanation of this stage)

    Again, multiply through:
    =3Sin(x)-3Sin^3(x)-Sin^3(x)

    Again, collect like terms:
    =3Sin(x)-4Sin^3(x)

    You'll have to check my working for silly errors, but the key thing to apply when you've reached the stage you have is just Cos^2(x)+Sin^2(x)=1 and rearrange that to convert everything into one trig. function - I used it at the indicated stage to convert the cosine function into a sine function.
    Last edited by Quacky; March 23rd 2011 at 04:53 PM.
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