Applying the Law of Cosines, the measure of the angles that the frontage will make with the two other boundaries are: 39.8° and 86.1°.
Recall:
- Law of Cosines is given as: c² = a² + b² - 2ab(Cos C)
The triangular parcel is shown in the diagram attached below.
a = 75 m
b = 117 m (frontage)
c = 95 m
∠A and ∠C represents the angles made with the two other boundaries.
Apply the Law of Cosines to solve for ∠A and ∠C.
Find ∠A using a² = c² + b² - 2cb(Cos A):
75² = 95² + 117² - 2(95)(117)(Cos A)
5,625 = 22,714 - 22,230(Cos A)
5,625 - 22,714 = -22,230(Cos A)
-17,089 = -22,230(Cos A)
- Divide both sides by -22,230
[tex]0.7687 = Cos A\\\\A = cos^{-1}(0.7687)\\\\A = 39.8[/tex]
Find ∠B using b² = a² + c² - 2ac(Cos B):
117² = 75² + 95² - 2(75)(95)(Cos B)
13,689 = 14,650 - 14,250(Cos B)
13,689 - 14,650 = -14,250(Cos B)
-961 = -14,250(Cos B)
- Divide both sides by -14,250
[tex]0.0674 = Cos B\\\\B = cos^{-1}(0.0674)\\\\B = 86.1[/tex]
Therefore, applying the Law of Cosines, the measure of the angles that the frontage will make with the two other boundaries are: 39.8° and 86.1°.
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