Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (read description)

Question

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers so that ∠B₁ is larger than ∠B₂.)

a= 32, c=48, angle A= 35 degrees

find,

Angle B1, angle C1, b1

Angle B2, C2, b2

Answer 1

Here we have a Side Side Angle Triangle. We are given side a= 32, side c=48, angle A= 35 degrees.

We will follow a process using 3 steps:

1. use The Law of Sines first to calculate one of the other two angles;
2. use the three angles add to 180° to find the other angle;
3. use The Law of Sines again to find the unknown side.

We can first use the Law of Sines to solve for the angle C:

sin(A)/a=sin(C)/c

sin(35)/32=sin(C)/48

Rearranging: sin(C)=(48*sin(35))/32)

Simplifying: sin(C)=0.86

Inverting: C=sin^-1(0.86)

Solving for C: C=59.35 degrees

Now understanding that A+B+C=180 degrees:

35+B+59.35=180

Rearranging: B=180-35-59.35

Solving for B: B=85.64 degrees

We can now use the Law of Sines to solve for the side b:

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

sin(35)/32=sin(85.64)/b

Rearranging: b=sin(85.64)/(sin(35)/32)

Solving for b: b=55.62

In this case, we assumed that angle C would be less than 90 degrees, but that is not always the case. There are two angles equivalent to sin^-1(0.86):

In order to calculate this angle, we subtract it from 180 degrees.

180-sin^-1(0.86)=180-59.35=120.65 degrees

C₂=120.65 degrees

We can now follow each of the previous steps to calculate the new triangle's values:

35+B+120.65=180

B=180-35-12.65=24.35

sin(35)/32=sin(24.35)/b

b=sin(24.35)/(sin(35)/32)

b=23.00

Answer 2

SSA - law of sine, given acA solve for C = sin^-1( 40 * sin( 35 ) / 32 ) = sin^-1( 0.71697) = 45.8°

SSA could be an ambiguous case! C could also be 134.2°

Since the sum of the three angles is 180, B = 180 - ( 35 + 45.8 ) = 99.2°

Using law of sine, given two angles and a side, b = (32 / sin( 35 ) ) * sin( 99.2 ) = 55.07