How do you solve the triangle when Angle A=37 degrees, a=49, b=54?

1 Answer
Nov 24, 2015

A=37°
a=49
B=arcsin((54sin(37°))/49)~~41.546274420887°
b=54
C=143°-arcsin((54sin(37°))/49)~~101.453725579113°
c=(49sin(143°-arcsin((54sin(37°))/49)))/sin(37°)~~79.7989130980759269

Explanation:

Make use of the Law of Sines:

a/sinA=b/sinB=c/sinC

Solving for B

Law of Sines

[1]" "a/sinA=b/sinB

Isolate sinB.

[2]" "cancela/cancelsinA*(cancel((sinA))(sinB))/cancela=b/cancelsinB*((sinA)cancel((sinB)))/a

[3]" "sinB=(bsinA)/a

Plug in the values of a, A, and b.

[4]" "sinB=(54sin(37°))/49

Get the inverse function. Solve using a calculator

[5]" "hArrcolor(blue)(B=arcsin((54sin(37°))/49)~~41.546274420887°)

Solving for C

The sum of all interior angles of a triangle is 180°.

[1]" "A+B+C=180°

Isolate C

[2]" "C=180°-A-B

Plug in the values of A and B.

[3]" "C=180°-37°-arcsin((54sin(37°))/49)

Solve using a calculator.

[4]" "color(blue)(C=143°-arcsin((54sin(37°))/49)~~101.453725579113°)

Solving for c

Law of Sines

[1]" "c/sinC=a/sinA

Isolate c.

[2]" "c/cancelsinC*cancelsinC=a/sinA*sinC

[3]" "c=(asinC)/sinA

Plug in the values of A, a, and C.

[4]" "c=(49sin(143°-arcsin((54sin(37°))/49)))/sin(37°)

Solve using a calculator.

[5]" "color(blue)(c=(49sin(143°-arcsin((54sin(37°))/49)))/sin(37°)~~79.7989130980759269)