trig

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write y=sinx-cosx in the form y=ksin (x+a), where the measure of a is in radians. answers: sqrt2sin (x+3pi/4), sqrt2sin (x+5pi/4), sqrt2sin (x+pi/4), or sqrt2sin (x+7pi/4) ...?

  • trig -

    ksin(x+a) = k(sinxcosa) + cosxsina)
    = ksinxcosa + kcosxsina

    then
    ksinxcosa + kcosxsina = sinx - cosx , which must be an identity, so true for all x
    let x =0
    k(0)cosa + k(1)sina = 0-1
    ksina= 1
    sina = 1/k

    let x = 90°
    k(1)(cosa) + k(0) = 1-0
    cosa = 1/k

    sina/cosa = (1/k) / (1/k)
    tana = 1
    a = 45° or π/4

    then is sina = 1/k
    √2/2 = 1/k
    k = 2/√2 = √2

    then
    y = sinx - cosx
    = √2sin(x+π/4)

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