Question

question 18 of 26
which of the following equations correctly represents the law of cosines?

a. ( b^2 = a^2 + b^2 - 2ab cdot cos(c) )
b. ( b^2 = a^2 + c^2 - 2ac cdot cos(b) )
c. ( c^2 = a^2 + b^2 - 2ab cdot cos(b) )
d. ( a^2 = b^2 + c^2 - 2ac cdot cos(a) )

Explanation:

Step1: Recall Law of Cosines

The law of cosines states that for a triangle with sides \(a\), \(b\), \(c\) and the angle opposite to side \(a\) is \(A\), opposite to \(b\) is \(B\), opposite to \(c\) is \(C\), the formulas are:
\(a^{2}=b^{2}+c^{2}-2bc\cos(A)\)
\(b^{2}=a^{2}+c^{2}-2ac\cos(B)\)
\(c^{2}=a^{2}+b^{2}-2ab\cos(C)\)

Step2: Analyze Option A

Option A: \(b^{2}=a^{2}+b^{2}-2ab\cdot\cos(C)\)
Subtract \(b^{2}\) from both sides: \(0 = a^{2}-2ab\cos(C)\), which is not the correct law of cosines form. So A is wrong.

Step3: Analyze Option B

Option B: \(b^{2}=a^{2}+c^{2}-2ac\cdot\cos(B)\)
This matches the law of cosines formula for \(b^{2}\) (since angle \(B\) is opposite side \(b\), and the formula uses the other two sides \(a\) and \(c\) and the cosine of angle \(B\)).

Step4: Analyze Option C

Option C: \(c^{2}=a^{2}+b^{2}-2ab\cdot\cos(B)\)
The correct formula for \(c^{2}\) should have \(\cos(C)\) (since angle \(C\) is opposite side \(c\)), so this is wrong.

Step5: Analyze Option D

Option D: \(a^{2}=b^{2}+c^{2}-2ac\cdot\cos(A)\)
The correct formula for \(a^{2}\) should have \(2bc\cos(A)\) (not \(2ac\cos(A)\)), so this is wrong.

Answer:

B. \(b^{2}=a^{2}+c^{2}-2ac\cdot\cos(B)\)