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]$c^2=a^2+b^2-2 a b \cos (C)$[/tex]
B. [tex]$c^2=a^2+b^2-2 b c \cos (B)$[/tex]
C. [tex]$c^2=a^2-b^2-2 b c \cos (C)$[/tex]
D. [tex]$c^2=a^2+b^2-2 b c \cos (A)$[/tex]
Answer (2)
The Law of Cosines relates the sides and angles of a triangle and is given by the formula c 2 = a 2 + b 2 − 2 ab cos ( C ) . Among the options provided, only option A correctly states this relationship. Therefore, the answer is A .
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The Law of Cosines relates the sides and angles in any triangle.
The formula is c 2 = a 2 + b 2 − 2 ab cos ( C ) , where C is the angle opposite side c .
Comparing the options, only option A matches the correct formula.
Therefore, the correct answer is A .
Explanation
Understanding the Law of Cosines The Law of Cosines relates the side lengths of a triangle to the cosine of one of its angles. It's a generalization of the Pythagorean theorem, which applies only to right triangles. The Law of Cosines is useful for finding the length of a side of a triangle if you know the lengths of the other two sides and the angle opposite the unknown side. It can also be used to find the angles of a triangle if you know the lengths of all three sides.
Recalling the Law of Cosines The standard form of the Law of Cosines is:
c 2 = a 2 + b 2 − 2 ab cos ( C )
Where:
a , b , and c are the side lengths of the triangle.
C is the angle opposite side c .
Comparing with the given options Now, let's compare the given options with the correct formula:
A. c 2 = a 2 + b 2 − 2 ab cos ( C ) - This matches the standard form of the Law of Cosines. B. c 2 = a 2 + b 2 − 2 b c cos ( B ) - This is incorrect because the last term should be − 2 ab cos ( C ) .
C. c 2 = a 2 − b 2 − 2 b c cos ( C ) - This is incorrect because the sign of b 2 is wrong. D. c 2 = a 2 + b 2 − 2 b c cos ( A ) - This is incorrect because the last term should be − 2 ab cos ( C ) .
Identifying the correct option The correct statement of the Law of Cosines is:
c 2 = a 2 + b 2 − 2 ab cos ( C )
Therefore, the correct option is A.
Examples
The Law of Cosines is incredibly useful in navigation and surveying. Imagine you're a surveyor trying to determine the distance across a lake. You can measure the distances from your position to two points on opposite sides of the lake, as well as the angle between those two lines of sight. Using the Law of Cosines, you can then calculate the distance across the lake without ever having to get wet! This principle is also used in GPS technology to calculate distances between satellites and receivers, helping you navigate your way around town.
