Question

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.

Answer 1

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