Given any triangle [tex]A B C[/tex] with corresponding side lengths [tex]a, b[/tex], and [tex]c[/tex], the law of cosines states:
A. [tex]b^2=a^2-c^2-2 b c \cos (B)[/tex]
B. [tex]b^2=a^2+c^2-2 b c \cos ^{(A)}[/tex]
C. [tex]b^2=a^2-c^2-2 b c \cos (C)[/tex]
D. [tex]B^2=a^2+c^2-2 a c \cos ^{(B)}[/tex]
Answer (2)
The law of cosines states b 2 = a 2 + c 2 − 2 a c cos ( B ) . Analyzing the provided options reveals that assuming option D contains a typo (where B^2 should be b^2), it closely reflects the correct formulation. Therefore, the best choice is option D.
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Recall the law of cosines: b 2 = a 2 + c 2 − 2 a c cos ( B ) .
Compare the formula with the given options.
Identify that option D is the closest to the correct form if we consider a typo where B 2 should be b 2 .
State the final answer: Assuming option D has a typo, the closest correct answer is D .
Explanation
Recall the Law of Cosines The law of cosines relates the side lengths of a triangle to the cosine of one of its angles. The standard formulation of the law of cosines for side b is:
Compare with the given options b 2 = a 2 + c 2 − 2 a c cos ( B ) Now, let's compare this with the given options:
Option A: b 2 = a 2 − c 2 − 2 b c cos ( B ) - Incorrect. Option B: b 2 = a 2 + c 2 − 2 b c cos ( A ) - Incorrect. Option C: b 2 = a 2 − c 2 − 2 b c cos ( C ) - Incorrect. Option D: B 2 = a 2 + c 2 − 2 a c cos ( B ) - Incorrect (capital B is used instead of b 2 on the left side).
Identify the correct option However, option D is closest to the correct form if we consider a typo where B 2 should be b 2 . The correct form should relate the square of a side to the squares of the other two sides and the cosine of the angle opposite to that side.
Correct the typo and choose the best option Considering the options, none of them are perfectly correct. However, option D is the closest if we assume there's a typo and B 2 should be b 2 . Then option D would be: b 2 = a 2 + c 2 − 2 a c cos ( B ) This matches the standard law of cosines.
State the final answer Therefore, assuming option D has a typo and should be b 2 instead of B 2 , option D is the correct answer.
Examples
The law of cosines is used in surveying to calculate distances and angles when direct measurement is not possible. For example, if you know the lengths of two sides of a triangular plot of land and the angle between them, you can use the law of cosines to find the length of the third side. This is particularly useful in situations where obstacles prevent direct measurement.
