Multiple choice

Construct a triangle with sides $AB = 4 cm, BC = 5 cm$ and $AC = 6 cm$ and then find $\angle{ACB}$ using cosine's rule

  1. $cos^{-1}{\dfrac{3}{4}}$
  2. $cos^{-1}{\dfrac{5}{8}}$
  3. $cos^{-1}{\dfrac{2}{4}}$
  4. $cos^{-1}{\dfrac{1}{4}}$
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A Correct answer
Explanation

Using the law of cosines: c^2 = a^2 + b^2 - 2ab cos(C). Here, c = AB = 4, a = BC = 5, b = AC = 6. 4^2 = 5^2 + 6^2 - 2(5)(6) cos(C). 16 = 25 + 36 - 60 cos(C). 60 cos(C) = 45. cos(C) = 45/60 = 3/4. C = cos^-1(3/4).

AI explanation

To find angle ACB, use the cosine rule which states that the cosine of the angle equals the sum of the squares of the adjacent sides minus the square of the opposite side, all divided by twice the product of the adjacent sides. The adjacent sides are AC equals 6 centimeters and BC equals 5 centimeters, while the opposite side AB equals 4 centimeters. Substituting these values gives the cosine of angle ACB as the quantity 36 plus 25 minus 16 divided by 2 times 6 times 5, resulting in 45 over 60. This fraction simplifies to 3 by 4, making the angle the inverse cosine of 3 by 4.

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