Determine the area of the triangle The figure is not to scal

Determine the area of the triangle. The figure is not to scale. The area is square centimeters. (Do not round until the final answer. Then round to two decimal places as needed.) es ra or Enter your answer in the answer box.

Solution

from figure

A = 57 , B = 77 , b = 10 cm

Area = 1/2 ( a b sin C ) or 1/2 ( b c sin A ) or 1/2 ( a c sin B )

Use the fact that the sum of all three angles of a triangle is equal to 180 ^o to write an equation in C

A + B + C = 180 ^o

57 + 77 + C = 180 ^o

C = 180 ^o - 57⁰ - 77⁰

C = 46⁰

Law of sines

sinA / a = sinB /b = sinC / c

==> sinA / a = sinB /b

==> a sin B = b sinA

==> a sin( 77) = 10 sin (57)

==> a = 10 ( sin( 77) / sin (57) )

==> a = 11.61803

So

A = (1/2) ( 11.61803 * 10 ) sin ( 46 )

A = (1/2) ( 11.61803 * 10 )( 0.71933980033)

A = 41.78657