Sure, let's analyze the given equations step-by-step to determine which one correctly uses the law of cosines.
The Law of Cosines states:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
We will compare each equation provided:
1. [tex]\(7^2=8^2+11^2-2(8)(11) \cos(N)\)[/tex]
2. [tex]\(8^2=7^2+11^2-2(7)(11) \cos(M)\)[/tex]
3. [tex]\(7^2=8^2+11^2-2(8)(11) \cos(P)\)[/tex]
4. [tex]\(8^2=7^2+11^2-2(7)(11) \cos(P)\)[/tex]
For each equation, we need to verify the correctness by calculating both sides and checking if they match.
Equation 1:
[tex]\[ 7^2 = 8^2 + 11^2 - 2(8)(11) \cos(N) \][/tex]
[tex]\[ 49 = 64 + 121 - 176 \cos(N) \][/tex]
Equation 2:
[tex]\[ 8^2 = 7^2 + 11^2 - 2(7)(11) \cos(M) \][/tex]
[tex]\[ 64 = 49 + 121 - 154 \cos(M) \][/tex]
Equation 3:
[tex]\[ 7^2 = 8^2 + 11^2 - 2(8)(11) \cos(P) \][/tex]
[tex]\[ 49 = 64 + 121 - 176 \cos(P) \][/tex]
Equation 4:
[tex]\[ 8^2 = 7^2 + 11^2 - 2(7)(11) \cos(P) \][/tex]
[tex]\[ 64 = 49 + 121 - 154 \cos(P) \][/tex]
None of these equations hold true for their respective sides; hence, none of the given equations correctly apply the law of cosines as presented.
This means none of the equations are correctly applying the law of cosines in this context.
