Wednesday, November 7, 2012

Practice problems similar to take home


a) Rewrite cos^4(alpha) - sin^4(alpha) in a simpler form with lower powers of the trig functions

b) let cosgamma = 5/7 find
singamma
tangamma
cos(½gamma)
sin(½gamma)
tan(½gamma)
cos(2gamma)
sin(2gamma)
tan(2gamma)

c) The sides of a triangle are A = 5, B = 5 and C = 8.

Find the area.
Find the approximate value of alpha in degrees, rounded to four places after the decimal.
Find the approximate value of beta in degrees, rounded to four places after the decimal.
Find the approximate value of gamma in degrees, rounded to four places after the decimal.
Find the height if A is the base.
Find the height if B is the base.
Find the height if C is the base.

d) A = 6, B = 7, beta = 90° Find sinalpha and alpha written in degrees rounded four places after the decimal.

Answers in the comments.
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1 comment:

  1. Prof. HubbardNovember 7, 2012 at 5:17 PM

    a) Rewrite cos^4(alpha) - sin^4(alpha) in a simpler form with lower powers of the trig functions.

    This is a difference of squares, so we can use

    (cos²alpha + sin²alpha)(cos²alpha - sin²alpha)

    The first factor is always 1, so the answer is simply cos²alpha - sin²alpha.


    b) let cosgamma = 5/7 find
    singamma = sqrt(1 - 25/49) = sqrt(24)/7 = 2sqrt(6)/7
    tangamma = 2sqrt(6)/5
    cos(½gamma) = sqrt(6/7) = sqrt(42)/7
    sin(½gamma) = sqrt(1/7) = sqrt(7)/7
    tan(½gamma) = sqrt(7/42) = sqrt(1/6) = sqrt(6)/6
    cos(2gamma) = 1/49
    sin(2gamma) = 20sqrt(6)/49
    tan(2gamma) = 20sqrt(6)

    c) The sides of a triangle are A = 5, B = 5 and C = 8.

    Find the area.
    S = ½(5+5+8) = 9
    Area = sqrt(9*4*4*1) = 12

    Find the approximate value of alpha in degrees, rounded to four places after the decimal.
    cos(alpha) = (5² + 8² - 5²)/(2*5*8) =
    64/80 = 4/5
    arccos(4/5) = 38.6498°


    Find the approximate value of beta in degrees, rounded to four places after the decimal.

    Because A = B, alpha = beta = 38.6498°
    Find the approximate value of gamma in degrees, rounded to four places after the decimal.
    The easiest way to get gamma is (180 - 2*38.6498)° = 102.6804°
    Find the height if A is the base.
    ½A*base_A = 12
    base_A = 24/5

    Find the height if B is the base.
    Same as base_A

    Find the height if C is the base.
    base_c = 6

    d) A = 6, B = 7, beta = 90° Find sinalpha and alpha written in degrees rounded four places after the decimal.

    sin(90°) = 1 so sinalpha/6 = 1/7
    sinalpha = 6/7
    arcsin(6/7) = 58.9973°

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