in triangle abc |BC|=4.12cm ,|AC|=6.82cm and angle BCA is 42.5 . Calculate|AB| . and angle bac and angle abc​

Answers

Answer 1

Answer:

i. |AB| = 4.70 cm

ii. BAC = [tex]36.3^{o}[/tex]

iii. ABC = [tex]101.2^{o}[/tex]

Explanation:

i Let |AB| be represented by c, |AC| by b and |BC| by a. Since the question would give an included angle triangle, we first apply cosine rule.

[tex]c^{2}[/tex] = [tex]a^{2}[/tex] + [tex]b^{2}[/tex] - 2ab Cos C

   = [tex](4.12)^{2}[/tex] + [tex](6.82)^{2}[/tex] - 2(4.12*6.82) Cos [tex]42.5^{o}[/tex]

   = 16.9744 + 46.5124 - 56.1968*0.7373

   = 63.4868 - 41.4340

[tex]c^{2}[/tex] = 22.0528

c = [tex]\sqrt{22.0528}[/tex]

  = 4.6960

c = 4.70 cm

Thus |AB| = 4.70 cm

ii. Applying the sine rule;

[tex]\frac{a}{sin A}[/tex] = [tex]\frac{b}{sin B}[/tex] = [tex]\frac{c}{sin C}[/tex]

[tex]\frac{4.12}{Sin A}[/tex] = [tex]\frac{4.7}{sin 42.5}[/tex]

sin A = [tex]\frac{4.12*sin 42.5}{4.7}[/tex]

         = 0.5922

A = [tex]sin^{-1}[/tex] 0.5922

   = 36.3

BAC = [tex]36.3^{o}[/tex]

But,

iii. BCA + BAC + ABC = [tex]180^{o}[/tex]

[tex]42.5^{o}[/tex] + [tex]36.3^{o}[/tex] + ABC = [tex]180^{o}[/tex]

ABC = [tex]180^{o}[/tex] - 78.8

       = 101.2

ABC = [tex]101.2^{o}[/tex]


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