Answer :
To determine which equation correctly applies the law of cosines to solve for an unknown angle measure, let’s go through each provided equation step-by-step and evaluate them.
The law of cosines is given by:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
where [tex]\( C \)[/tex] is the angle opposite side [tex]\( c \)[/tex], and [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the other two sides.
### Equation 1:
[tex]\[ 7^2 = 8^2 + 11^2 - 2(8)(11) \cos (N) \][/tex]
For the equation to hold true according to the law of cosines, we need to verify if:
[tex]\[ 7^2 = 8^2 + 11^2 - 2(8)(11) \cos(N) \][/tex]
### Equation 2:
[tex]\[ 8^2 = 7^2 + 11^2 - 2(7)(11) \cos (M) \][/tex]
For the equation to hold true according to the law of cosines, we need to verify if:
[tex]\[ 8^2 = 7^2 + 11^2 - 2(7)(11) \cos(M) \][/tex]
### Equation 3:
[tex]\[ 7^2 = 8^2 + 11^2 - 2(8)(11) \cos (P) \][/tex]
For the equation to hold true according to the law of cosines, we need to verify if:
[tex]\[ 7^2 = 8^2 + 11^2 - 2(8)(11) \cos(P) \][/tex]
This equation is exactly the same as Equation 1, so they should both be either correct or incorrect together.
### Equation 4:
[tex]\[ 8^2 = 7^2 + 11^2 - 2(7)(11) \cos (P) \][/tex]
For the equation to hold true according to the law of cosines, we need to verify if:
[tex]\[ 8^2 = 7^2 + 11^2 - 2(7)(11) \cos(P) \][/tex]
Analyzing all these equations, none of them satisfy the conditions of the law of cosines based on the known relationships and the given results.
Hence, the final answer is that none of these equations correctly applies the law of cosines to solve for an unknown angle measure:
[tex]\[ \boxed{\text{None}} \][/tex]
The law of cosines is given by:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
where [tex]\( C \)[/tex] is the angle opposite side [tex]\( c \)[/tex], and [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the other two sides.
### Equation 1:
[tex]\[ 7^2 = 8^2 + 11^2 - 2(8)(11) \cos (N) \][/tex]
For the equation to hold true according to the law of cosines, we need to verify if:
[tex]\[ 7^2 = 8^2 + 11^2 - 2(8)(11) \cos(N) \][/tex]
### Equation 2:
[tex]\[ 8^2 = 7^2 + 11^2 - 2(7)(11) \cos (M) \][/tex]
For the equation to hold true according to the law of cosines, we need to verify if:
[tex]\[ 8^2 = 7^2 + 11^2 - 2(7)(11) \cos(M) \][/tex]
### Equation 3:
[tex]\[ 7^2 = 8^2 + 11^2 - 2(8)(11) \cos (P) \][/tex]
For the equation to hold true according to the law of cosines, we need to verify if:
[tex]\[ 7^2 = 8^2 + 11^2 - 2(8)(11) \cos(P) \][/tex]
This equation is exactly the same as Equation 1, so they should both be either correct or incorrect together.
### Equation 4:
[tex]\[ 8^2 = 7^2 + 11^2 - 2(7)(11) \cos (P) \][/tex]
For the equation to hold true according to the law of cosines, we need to verify if:
[tex]\[ 8^2 = 7^2 + 11^2 - 2(7)(11) \cos(P) \][/tex]
Analyzing all these equations, none of them satisfy the conditions of the law of cosines based on the known relationships and the given results.
Hence, the final answer is that none of these equations correctly applies the law of cosines to solve for an unknown angle measure:
[tex]\[ \boxed{\text{None}} \][/tex]
