PreCalc

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If tan theta = 9/5 and cot omega = 9/5
Find the exact value of sin (omega-theta)

  • PreCalc -

    sorry that should be theta-omega not omega-theta

  • PreCalc -

    Cleverly, you note that since
    tanθ = cotω, ω = π/2-θ
    Thus, θ-ω = θ-(π/2-θ) = π/2+2θ
    sin(θ-ω) = -cos2θ = 2sin^θ-1
    tanθ = 9/5, so
    sinθ = 9/√106
    sin(θ-ω) = 162/106 - 1 = 56/106

    or, if you must exercise your sum/difference formulas,

    sin(θ-ω) = sinθcosω-cosθsinω

    tanθ = 9/5, so
    sinθ = 9/√106
    cosθ = 5/√106

    cotω = 9/5, so
    sinω = 5/√106
    cosω = 9/√106

    sin(θ-ω) = 9/√106 * 9/√106 - 5/√106 * 5/√106 = 56/106

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