Posted by Britt on Friday, December 16, 2011 at 12:41am.
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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