Solve for unknown

Question

Solve for the unknown, show all steps. Round to 1 decimal place.

(a) sin50°/x = sin60°/8.0

(b) 36² = 32² + 40² − 2(32)(40)cosA

Answer 1

(b) is the formula for the law of cosines

a^2 = c^2 + b^2 -2cbcosA where A is the angle opposite side a

1296 = 1024 + 1600 - 2560cosA

1328 = 2560cosA

cosA - 1328/2560 = 0.51876

A= about 58.752 = 58.8 degrees

(a) is the law of sines

sin50/x = sin60/8 = sqr3/2/8 = sqr3/16

x= 16sin50/sqr3 = about 7.077 = 7.1

the ratio of sine of an angle of a triangle to the side opposite the angle is the same for each angle and its opposite side. sinA/a = sinB/b = sinC/c

sinA/a = sinB/b. sin60>sin50, so 8>x. sin60 is little larger than sin50, so 8 is a little larger than x