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trig

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How do you find the exact value of sin 5pi/8?

  • trig -

    cos2θ = 1 - 2sin^2 θ
    since 2θ = 5pi/4, cos 2θ = -1/√2

    -1/√2 = 1 - 2sin^2 5π/8
    2sin^2 5π/8 = 1 + 1/√2
    sin 5π/8 = √(1 + 1/√2)/2
    = √(2+√2) / 2

  • trig -

    You are talking about 112.5 degrees
    that is 90 + 22.5

    22.5 is half of 45

    so draw this in the third quadrant
    I know functions of 45 degrees
    sin 45 = +1/sqrt 2
    cos 45 = +1/sqrt 2
    so what are the functions of 22.5 degrees?
    sin(45/2) = +/- sqrt[(1-cos 45)/2]
    cos (45/2) = +/-sqrt[(1+cos 45)/2]
    in this case we want the negative cos for the sin (look at sketch) because in quadrant 3
    so sin 112.5 = -sqrt [ (1+1/sqrt 2)/2]
    = -sqrt [ (1 + sqrt 2)/2 sqrt 2
    =- sqrt [ (2+sqrt2)/4 ]
    = - .923
    check sin 112.5 on calculator = -.923

  • trig - note -

    sin is positive in QII

  • trig -

    whoops, sorry

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