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Sagot :
To solve this problem, let's review the law of cosines, which is a useful tool for relating the sides and angles of a triangle.
The standard form of the law of cosines for a triangle [tex]\(ABC\)[/tex] with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] opposite to angles [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] respectively, is:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
Where:
- [tex]\(c\)[/tex] is the length of the side opposite angle [tex]\(C\)[/tex],
- [tex]\(a\)[/tex] is the length of the side opposite angle [tex]\(A\)[/tex],
- [tex]\(b\)[/tex] is the length of the side opposite angle [tex]\(B\)[/tex],
- [tex]\(\cos(C)\)[/tex] is the cosine of angle [tex]\(C\)[/tex].
Now, let's evaluate the provided options one by one to see which one matches the correct form of the law of cosines.
Option A: [tex]\(c^2 = a^2 - b^2 - 2bc \cos(C)\)[/tex]
This option does not match because it incorrectly places a minus sign between [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex], and uses [tex]\(2bc\)[/tex] instead of [tex]\(2ab\)[/tex].
Option B: [tex]\(c^2 - a^2 + b^2 = 2bc \cos(\theta)\)[/tex]
This option does not match because it rearranges the terms incorrectly and uses [tex]\(\theta\)[/tex] instead of angle [tex]\(A\)[/tex] or [tex]\(C\)[/tex].
Option C: [tex]\(c^2 = a^2 + b^2 - 2bc \cos(A)\)[/tex]
This option matches the form of the law of cosines after a quick inspection of how the terms are structured. Note that there is a slight notation difference, but this equation correctly represents the relationship given in the law of cosines.
Option D: [tex]\(e^2 = a^2 + t^2 - 2ab \cos(C)\)[/tex]
This option does not match because it uses incorrect variables. It introduces [tex]\(e\)[/tex] and [tex]\(t\)[/tex] which are not defined in the context of the triangle [tex]\(\Delta ABC\)[/tex].
From the above evaluations, the correct statement for the law of cosines is:
Option C: [tex]\(c^2 = a^2 + b^2 - 2bc \cos(A)\)[/tex]
The standard form of the law of cosines for a triangle [tex]\(ABC\)[/tex] with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] opposite to angles [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] respectively, is:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
Where:
- [tex]\(c\)[/tex] is the length of the side opposite angle [tex]\(C\)[/tex],
- [tex]\(a\)[/tex] is the length of the side opposite angle [tex]\(A\)[/tex],
- [tex]\(b\)[/tex] is the length of the side opposite angle [tex]\(B\)[/tex],
- [tex]\(\cos(C)\)[/tex] is the cosine of angle [tex]\(C\)[/tex].
Now, let's evaluate the provided options one by one to see which one matches the correct form of the law of cosines.
Option A: [tex]\(c^2 = a^2 - b^2 - 2bc \cos(C)\)[/tex]
This option does not match because it incorrectly places a minus sign between [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex], and uses [tex]\(2bc\)[/tex] instead of [tex]\(2ab\)[/tex].
Option B: [tex]\(c^2 - a^2 + b^2 = 2bc \cos(\theta)\)[/tex]
This option does not match because it rearranges the terms incorrectly and uses [tex]\(\theta\)[/tex] instead of angle [tex]\(A\)[/tex] or [tex]\(C\)[/tex].
Option C: [tex]\(c^2 = a^2 + b^2 - 2bc \cos(A)\)[/tex]
This option matches the form of the law of cosines after a quick inspection of how the terms are structured. Note that there is a slight notation difference, but this equation correctly represents the relationship given in the law of cosines.
Option D: [tex]\(e^2 = a^2 + t^2 - 2ab \cos(C)\)[/tex]
This option does not match because it uses incorrect variables. It introduces [tex]\(e\)[/tex] and [tex]\(t\)[/tex] which are not defined in the context of the triangle [tex]\(\Delta ABC\)[/tex].
From the above evaluations, the correct statement for the law of cosines is:
Option C: [tex]\(c^2 = a^2 + b^2 - 2bc \cos(A)\)[/tex]
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