Question

in ABC, A = 61°, a = 25, and b = 27. What are the two possible values for angle B to the nearest tenth of a degree?

Answer 1

In ΔABC

- a, b, c are the lengths of its 3 sides, where

# a is opposite to angle A

# b is opposite to angle B

# c is opposite to angle C

- m∠A = 61°

a = 25

and

b = 27

To find m∠B we can use the sin Rule:

`(b)/(\sin(B))=(a)/(\sin (A))`

replacing the data of the problem in the previous equation, we get:

`(27)/(\sin (B))=(25)/(\sin (61))`

by cross-multiplication, we get:

`\sin (B)(25)=\text{ sin(61)(27)}`

solving for sin(B), we get:

`\sin (B)=(\sin (61)(27))/(25)=0.94`

applying the inverse function of sine, we get:

`B=sin^(-1)(0.94)=70.8`

note that the value of sin(B) is positive

∴ Angle B may be in the first quadrant (acute angle) or in the second quadrant (obtuse angle). Thus, the other measure of ∠B would be:

`B\text{ = }180-70.8=\text{ 109}.2`

Then, the two possible values of B are:

`B\text{ = 109}.2`

and

`B\text{ = 70.8}`