Math

1) use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle.

67 degrees and 30'

Please explain!! I do not know how to start.

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  1. Notice that 67 degrees, 30' is half of 135 degrees,
    and 135 = 180-45

    We also know that
    cos 2A = 2 cos^2 A - 1
    giving us
    cos 135 = 2cos^2 67.5 - 1

    So let's find cos 135
    135 is in quadrant II, so
    cos 135 = -sin 45 = -1/√2

    back to cos 135 = 2cos^2 67.5 - 1
    -1/√2 = 2cos^2 67.5 - 1
    1 - 1/√2 = 2cos^2 67.5
    (√2-1)/(2√2) = cos^2 67.5
    cos 67.5 = √[(√2-1)/(2√2)]
    (my calculator confirmed my answer to be correct

    Using the other version of
    cos 2A = 1 - 2sin^2 A -1 and following the above steps will give you the sin 67.5

    you then find tan 67.5 by sin67.5/cos67.5 using the expressions.

    Enjoy the algebra, lol

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