To determine which of the given equations correctly uses the law of cosines to solve for [tex]\( y \)[/tex], we need to carefully examine each equation and ensure it aligns with the standard form of the law of cosines.
The law of cosines states:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
In this context, we can match:
- [tex]\( a \)[/tex] with [tex]\( y \)[/tex], which is the side we need to solve for.
- [tex]\( b \)[/tex] with [tex]\( 9 \)[/tex]
- [tex]\( c \)[/tex] with [tex]\( 19 \)[/tex]
- [tex]\( A \)[/tex] with [tex]\( 41^\circ \)[/tex], which is the angle opposite side [tex]\( y \)[/tex].
Let's rewrite each option to see if it matches the form [tex]\( a^2 = b^2 + c^2 - 2bc \cos(A) \)[/tex].
1. [tex]\( 9^2 = y^2 + 19^2 - 2(y)(19) \cos(41^\circ) \)[/tex]
This equation is incorrect because in the law of cosines, [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex] should all be on one side of the equation. Here, the equation has the variables mixed up across both sides.
2. [tex]\( y^2 = 9^2 + 19^2 - 2(y)(19) \cos(41^\circ) \)[/tex]
This equation is incorrect. The term [tex]\( 2(y)(19) \cos(41^\circ) \)[/tex] should involve the sides we know the lengths of, not the side we are solving for (i.e., it should involve 9 and 19, not [tex]\(y\)[/tex]).
3. [tex]\( 9^2 = y^2 + 19^2 - 2(9)(19) \cos(41^\circ) \)[/tex]
This equation is incorrect because it incorrectly places [tex]\( y^2 \)[/tex] with the known side lengths [tex]\( 9 \)[/tex] and [tex]\( 19 \)[/tex] on the right-hand side of the equation.
4. [tex]\( y^2 = 9^2 + 19^2 - 2(9)(19) \cos(41^\circ) \)[/tex]
This equation correctly follows the structure of the law of cosines:
[tex]\[
y^2 = 9^2 + 19^2 - 2 \cdot 9 \cdot 19 \cdot \cos(41^\circ)
\][/tex]
This is in the form:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
where [tex]\( a = y \)[/tex], [tex]\( b = 9 \)[/tex], [tex]\( c = 19 \)[/tex], and [tex]\( A = 41^\circ \)[/tex].
Thus, the correct equation that uses the law of cosines to solve for [tex]\( y \)[/tex] is:
[tex]\[ y^2 = 9^2 + 19^2 - 2(9)(19) \cos(41^\circ) \][/tex]
So, the correct answer is:
[tex]\[ y^2 = 9^2 + 19^2 - 2(9)(19) \cos(41^\circ) \][/tex]
