The measures of two sides of a triangle and the angle opposite one of them are given. Solve all possible triangles if any. (With solution and explanation pls)

Question

a) c = 9.04, a = 3.52, γ = 128.1°

b) b = 3.562, c = 4.210, β = 50.23°

Answer 1

two triangles with the given info, for part b)

one scalene with all acute angles

A,B,C 64.479, 50.23, 65.291 degrees

a,b, c 4.182092791, 3.562, 4.21

but there's a 2nd triangle possible, obtuse with one angle >90 degrees

A,B,C 15.061, 50.23, 114.709 degrees

a,b,c 1.204, 3.502, 4.21

use law of sines sinC/c = sinB/b, c=4.21, b =3.562, B = 50.23

part b)

sinC/4.21 = sin50.23/3.562

sinC = 4.21sin50.23/3.562

C= arcsin(4.21sin50.23/3.562)= arcsin.908

C= about 65.3 or 114.7degrees

A=180-B-C = 180-50.23-65.3 or 180-50.23-114.7

A =65.3 or 114.7 degrees

using Law of Cosines

a^2= b^2 +c^2- 2bcCosA

a = square root of (b^2+c^2-2bcCosA)

try sketching the two triangles

there were 3 possibilities, 0 triangle, 1 triangle 2 triangles

in this problem turns out there are 2 triangles that fit the given b,c,B sides and angle

then there is part a)

c=9.04, a=3.52, angle gamma or B= 128.1, opposite side b

A=13.87751091 degrees opposite side a

C= 38.02248909 degrees opposite side c

b= about 11.54908464 opposite angle B or gamma, the obtuse angle

then is that it? or is there a 2nd triangle for part a)?

first impression is that's it, just one triangle

math or trigonometry uses greek letters, for angles

alpha for A, beta for B, gamma for C. there is no Greek letter similar to English C,

gamma is the 3rd letter of the Greek alphabet, like C in the English alphabet

Answer 2

Use the Law of sines and arcsin to determine the measure of angle A

A = arcsin( a*sin(C)/c ).

Use the Triangle Angle Identity to find angle B

B = 180 - A - C

Use the law of cosines to find the length of side b

b = sqrt( a² + c² - 2ac cos(B) )

Solve these values and you have completed the triangle.