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Math Help - Combination of sinusoidal functions

  1. #1
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    Combination of sinusoidal functions

    The problem is:
    v(t) = 5 sin (1000t +30 degrees)
    u(t) = 12 cos (1000t + 30 degrees)
    solve for v(t) + u(t).

    The 12 cos (1000t + 30) was converted to 12 sin (1000t - 60 degrees)

    I converted to polar notation and converted using Euler's equation. The two functions were then summed to form:

    10.33 + j7.9

    The magnitude was calculated using Pythagorean's theorem, and was 13.

    When I tried to calculate the angle using atan(7.9 / 10.33), I arrived at an angle of 37.4 degrees. The answer key indicates that it should have been 97.4 degrees.

    Can anyone tell me where I went wrong?
    Thank you.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by mattbrrtt View Post
    The problem is:
    v(t) = 5 sin (1000t +30 degrees)
    u(t) = 12 cos (1000t + 30 degrees)
    solve for v(t) + u(t).

    The 12 cos (1000t + 30) was converted to 12 sin (1000t - 60 degrees)

    I converted to polar notation and converted using Euler's equation. The two functions were then summed to form:

    10.33 + j7.9

    The magnitude was calculated using Pythagorean's theorem, and was 13.

    When I tried to calculate the angle using atan(7.9 / 10.33), I arrived at an angle of 37.4 degrees. The answer key indicates that it should have been 97.4 degrees.

    Can anyone tell me where I went wrong?
    Thank you.
    Poorly expressed problem, does not make explicit what it wants as a solution, but lets suppose it wants the signal in the form s(t)=\sin(\omega t+ \phi)

    s(t)=u(t)+v(t)=5 \sin (1000t +30)+12 \cos (1000t + 30)

    put \theta=\text{atan}(12/5), then:

    s(t)=13 [\cos(\theta) \sin(1000t+30)+\sin(\theta)\cos(1000t+30)]

    then its just trig.

    CB
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  3. #3
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    Baltimore, MD
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    Thank you

    I should have provided more information. The topic is sinusoidal stead state analysis. The two functions u(t) and v(t) are two signals of different vectors. The question was to determine the sum of the two signals in the form of a single sinusoidal function.
    The mistake with my equation was an incorrect translation in the angle that was changed prior to the conversion to polar form. The sum should have been -1.67 + j12.89. The degree, arctan(12.89/1.67) = 82.62 degrees, and the magnitude equals sqrt{(12.89^2)+(1.67^2)} = 13. Since the vector is in the second quadrant, the vector angle = 180 - 82.6 = 97.4 degrees.
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