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I have spent hours trying to do these, help!

Use the given information to solve the triangle.


1) C = 145 degrees, b = 4, c = 14.



2) A = 150 degrees, C = 20 degrees, a = 200.

  • math -

    1)

    Law of cosines:

    a ^ 2 = b ^ 2 + c ^ 2 - 2 b c cos( A )

    b ^ 2 = a ^ 2 + c ^ 2 - 2 a c cos( B )

    c ^ 2 = a ^ 2 + b ^ 2 - 2 a b cos( C )



    C = 145°, c = 14, b = 4

    c ^ 2 = a ^ 2 + b ^ 2 - 2 a b cos( C )

    14 ^ 2 = a ^ 2 + 4 ^ 2 - 2 a *4 *cos (C )

    196 = a ^ 2 + 16 - 8 a *cos ( 145° )

    196 - 16 = a ^ 2 - 8 a * cos ( 145° )

    180 = a ^ 2 - 8 a * cos( 145° )

    a ^ 2 - 8 a * cos( 145° ) - 180 = 0

    cos ( 145 ° ) = - cos ( 35° ) = -0.81915

    a ^ 2 - 8 a * ( -0.81915 ) - 180 = 0

    a ^ 2 + 6,5532 a - 180 = 0

    The solutions of this quadratic equation are :

    a = - 17.0873

    and

    a = 10.5341

    Length can't be negative so:

    a = 10.5341


    Law of Sines:

    sin ( A ) / a = sin ( B ) / b = sin ( C ) / c


    sin ( A ) / a = sin ( C ) / c Multiply both sides with a

    sin ( A ) = a * sin ( C ) / c

    sin ( A ) = 10.5341 * sin ( 145° ) / 14


    sin ( 145° ) = sin ( 35° ) = 0.57357


    sin ( A ) = 10.5341 * 0.57357 / 14

    sin ( A ) = 6,042043737 / 14

    sin ( A ) = 0,43157

    A = 25° 34´


    sin ( B ) / b = sin ( C ) / c Multiply both sides with b

    sin ( B ) = b * sin ( C ) / c

    sin ( B ) = 4 * 0.57357 / 14

    sin ( B ) = 2.29428 / 14

    sin ( B ) = 0.16388

    B = 9 ° 26´


    2)

    B = 180° - 150° - 20 °

    B = 10°


    Law of Sines:

    sin ( A ) / a = sin ( B ) / b = sin ( C ) / c


    sin ( A ) / a = sin ( B ) / b Multiply both sides with b

    b * sin ( A ) / a = sin ( B ) Divide both sides with sin ( A )

    b / a = sin ( B ) / sin ( A ) Multiply both sides with a

    b = a * sin ( B ) / sin ( A )

    b = 200 * sin ( 10° ) / sin ( 150° )

    sin ( 10° ) = 0.17365

    sin ( 150° ) = sin ( 30°) = 0.5


    b = 200 * sin ( 10° ) / sin ( 150° )

    b = 200 * 0.17365 / 0.5

    b = 34.73 / 0.5 = 34.73 * 2 = 69.46

    Remark: 1 / 0.5 = 2


    sin ( A ) / a = sin ( C ) / c Multiply both sides with c

    c * sin ( A ) / a = sin ( C ) Divide both sides with sin ( A )

    c / a = sin ( C ) / sin ( A ) Multiply both sides with a

    c = a * sin ( C ) / sin ( A )

    c = 200 * sin ( 20° ) / sin ( 150° )

    sin ( 20° ) = 0.34202

    c = 200 * sin ( 20° ) / sin ( 150° )

    c = 200 * 0.34202 / 0.5

    c = 68.404 / 0.5 = 68.404 * 2 = 136.808

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