Answer:
From figure A
The value of ∠ B = 75.74° , ∠ C = 70.26° and AB = 27.37
From figure B
The value of ∠ A = 42.8° , ∠ B = 106.2° and AC = 30.04
Step-by-step explanation:
Given first figure as :
AC = 28.2
BC = 16.5
∠ A = 34°
Let AB = c
From law of sines
[tex]\dfrac{a}{Sin A}[/tex] = [tex]\dfrac{b}{Sin B}[/tex] = [tex]\dfrac{c}{Sin C}[/tex]
Or, [tex]\dfrac{a}{Sin A}[/tex] = [tex]\dfrac{b}{Sin B}[/tex]
or, [tex]\dfrac{16.5}{Sin 34}[/tex] = [tex]\dfrac{28.2}{Sin B}[/tex]
Or, 29.506 = [tex]\dfrac{28.2}{Sin B}[/tex]
Or, Sin B = [tex]\dfrac{28.2}{29.5}[/tex]
Or, Sin B = 0.955
∴ ∠B = [tex]Sin^{-1}[/tex] 0.955
I.e∠ B = 75.74
Now, ∠ C = 180° - ( ∠A + ∠B )
Or, ∠ C = 180° - ( 34° + 75.74° )
Or, ∠ C = 70.26°
Now, Again
[tex]\dfrac{b}{Sin B}[/tex] = [tex]\dfrac{c}{Sin C}[/tex]
so, [tex]\dfrac{28.2}{Sin 75.74}[/tex] = [tex]\dfrac{c}{Sin 70.26}[/tex]
Or, [tex]\dfrac{28.2}{0.9691}[/tex] = [tex]\dfrac{c}{0.9412}[/tex]
Or, c = 29.09 × 0.9412
∴ c = 27.37
I.e AB = 27.37
Hence, The value of ∠ B = 75.74° , ∠ C = 70.26° and AB = 27.37
From figure second
Given as :
AB = c= 12
BC = a = 16
∠ C = 31°
let AC = b
From law of sines
[tex]\dfrac{a}{Sin A}[/tex] = [tex]\dfrac{b}{Sin B}[/tex] = [tex]\dfrac{c}{Sin C}[/tex]
Or, [tex]\dfrac{a}{SinA }[/tex] = [tex]\dfrac{c}{Sin C}[/tex]
or, [tex]\dfrac{16}{Sin A}[/tex] = [tex]\dfrac{12}{Sin 31}[/tex]
or, [tex]\dfrac{16}{Sin A}[/tex] = [tex]\dfrac{12}{0.51}[/tex]
Or, [tex]\dfrac{16}{Sin A}[/tex] = 23.52
∴ Sin A = [tex]\dfrac{16}{23.52}[/tex]
I.e Sin A = 0.68
Or, ∠ A = [tex]Sin^{-1}[/tex] 0.68
or, ∠ A = 42.8°
Now, ∠ B = 180° - ( 31° + 42.8° )
Or, ∠ B = 106.2°
Now, [tex]\dfrac{b}{Sin B}[/tex] = [tex]\dfrac{c}{Sin C}[/tex]
or, [tex]\dfrac{b}{Sin 106.2}[/tex] = [tex]\dfrac{16}{Sin 31}[/tex]
Or, [tex]\dfrac{b}{0.96}[/tex] = [tex]\dfrac{16}{0.51}[/tex]
or, b = 31.3×0.96
∴ b = 30.04
Hence The value of ∠ A = 42.8° , ∠ B = 106.2° and AC = 30.04
Answer
