The Law of Cosines
1 Use the Law of Cosines to Solve Oblique Triangles
-$$a = 4 , c = 2 , B = 90 ^ { \circ }$$
A) $\mathrm { b } = 4.5 , \mathrm {~A} = 63 ^ { \circ } , \mathrm { C } = 27 ^ { \circ }$
B) $\mathrm { b } = 4.5 , \mathrm {~A} = 27 ^ { \circ } , \mathrm { C } = 63 ^ { \circ }$
C) $\mathrm { b } = 5.5 , \mathrm {~A} = 63 ^ { \circ } , \mathrm { C } = 27 ^ { \circ }$
D) $b = 3.5 , \mathrm {~A} = 27 ^ { \circ } , \mathrm { C } = 63 ^ { \circ }$

Question

Quizplus · Accepted Answer

The Law of Cosines states that in a triangle with sides a, b, and c, and angles A, B, and C, the following equation holds:$$c^2 = a^2 + b^2 - 2ab \cos{C}$$In this case, we have a = 4, c = 2, and B = 90 degrees. We can use the Law of Cosines to solve for b:$$2^2 = 4^2 + b^2 - 2(4)(b) \cos{90}$$$$4 = 16 + b^2 - 8b$$$$b^2 - 8b + 12 = 0$$$$(b - 6)(b - 2) = 0$$$$b = 6, 2$$Since b cannot be negative, we have b = 6.We can also use the Law of Cosines to solve for A:$$A = \cos^{-1}{\frac{a^2 + c^2 - b^2}{2ac}}$$$$A = \cos^{-1}{\frac{4^2 + 2^2 - 6^2}{2(4)(2)}}$$$$A = \cos^{-1}{\frac{4}{16}}$$$$A = 63 degrees$$Finally, we can use the fact that the sum of the angles in a triangle is 180 degrees to solve for C:$$C = 180 - A - B$$$$C = 180 - 63 - 90$$$$C = 27 degrees$$Therefore, the correct answer is A) b = 4.5, A = 63 degrees, C = 27 degrees.

The Law of Cosines 1 Use the Law of Cosines a=4,c=2,B=90∘a = 4 , c = 2 , B = 90 ^ { \circ }a=4,c=2,B=90∘

The Law of Cosines
1 Use the Law of Cosines $a = 4 , c = 2 , B = 90 ^ { \circ }$