12th grade Trigonometry

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Solve this trigonometric equation.give all positive values of the angle between 0deg and 360deg that will satisfy.
Give any approximate value to the nearest minute only.

3 sin theta - 4 cos theta = 2

  • 12th grade Trigonometry -

    theta = 209,55192°
    theta = 76,70829°

  • 12th grade Trigonometry -

    Sum or difference of sin(θ) and cos(θ) can be solved in the following way, if you are familiar with the sum and difference formulae for sin(a±b).

    Given
    3sin(θ)-4cos(θ)=2
    for a particular φ and amplitude A, we rewrite the above equation as:
    Asin(θ-φ)=2
    where A=sqrt(3²+4²)=5
    or
    sin(θ-&phi)=2/5....(1)
    => θ-&phi=asin(2/5)=23.5782°
    or 180-23.5782=156.4218°. ...(2)

    Expand (1) to get:
    sin(θ)cos(&phi)-cos(θ)sin(&phi)=2/5
    => cos(φ)=3/5, sin(φ)=4/5
    => φ=53.1301° or 306.8699°...(3)

    So solve for θ from (2) and (3) gives
    θ=23.5782+53.1301=76.7083° or θ=156.4218+53.1301=209.5519°

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