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Math Help - [SOLVED] Sine/Cosine Addition Formulae

  1. #1
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    [SOLVED] Sine/Cosine Addition Formulae

    Express x = 2sin(0.5t + 3) + 3cos(0.5t + 1) in the form x = Rsin(0.5t + °)

    All working must be done in radians.

    I am stuck trying to solve this question.

    I understand that:

    asinx + bcosx can be written as Rsin(x + °)

    Where: Tan° = b/a and: R = sqrt(a▓ + b▓)

    The values for x (the frequency) in the question however have different numbers added to them. Does anyone know how to get around this? Thanks
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  2. #2
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    Quote Originally Posted by MathsDude69 View Post
    Express x = 2sin(0.5t + 3) + 3cos(0.5t + 1) in the form x = Rsin(0.5t + °)

    All working must be done in radians.

    I am stuck trying to solve this question.

    I understand that:

    asinx + bcosx can be written as Rsin(x + °)

    Where: Tan° = b/a and: R = sqrt(a▓ + b▓)

    The values for x (the frequency) in the question however have different numbers added to them. Does anyone know how to get around this? Thanks
    Hi

    You can expand

    2 \sin(0.5t+3) = 2 \cos 3 \sin 0.5t + 2 \sin 3 \cos 0.5t

    3 \cos(0.5t+1) = 3 \cos 1 \cos 0.5t - 3 \sin 1 \sin 0.5t

    and sum

    2 \sin(0.5t + 3) + 3 \cos(0.5t + 1) = (2 \cos 3 - 3 \sin 1) \sin 0.5t + (2 \sin 3 + 3 \cos 1) \cos 0.5t

    and then use the way you have written
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  3. #3
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    Hey:

    So here's my attempt. Can someone please check whether ive gone completely wrong with this:

    asinx + bcosx can be written as X = Rsin(x + °)

    Where: Tan° = b/a and: R = sqrt(a▓ + b▓)

    Expansion of terms =
    2sin(0.5t + 3) = 2cos3sin0.5t + 2sin3cos0.5t
    3cos(0.5t + 1) = 3cos1cos0.5t - 3sin1sin0.5t

    Addition of terms =
    2sin(0.5t + 3) + 3cos(0.5t + 1) = (2cos3-3sin1)sin0.5t + (2sin3+3cos1)cos0.5t

    Tan° = b/a

    Tan° = 3/2

    Tan° = 1.5

    ° = Tan^-1(1.5)

    ° = 0.983

    R = sqrt(a^2 + b^2)

    R = sqrt(4+9)

    R = sqrt (13)

    R = 3.606

    Therefore x = 3.606sin(0.5t + 1.5)


    Is this right or just completely wrong?? lol
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  4. #4
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    Addition of terms =
    2sin(0.5t + 3) + 3cos(0.5t + 1) = (2cos3-3sin1)sin0.5t + (2sin3+3cos1)cos0.5t

    Tan° = b/a

    Tan° = (2sin3+3cos1) / (2cos3-3sin1)


    R = sqrt(a^2 + b^2)

    R = sqrt((2cos3-3sin1)▓+(2sin3+3cos1)▓)

    R = sqrt (13 - 12 cos3 sin1 + 12 sin3 cos 1)

    R = sqrt (13 + 12 sin2)
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