To determine which equation correctly uses the Law of Cosines to solve for [tex]\( y \)[/tex], we need to carefully analyze each given option and compare it to the general form of the Law of Cosines. The Law of Cosines states:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
Here, [tex]\( a \)[/tex] is the side opposite angle [tex]\( A \)[/tex], and [tex]\( b \)[/tex] and [tex]\( c \)[/tex] are the other two sides of the triangle.
Given the problem, let's define the sides and the angle:
- [tex]\(a = 9\)[/tex]
- [tex]\(b = 19\)[/tex]
- [tex]\(\theta = 41^\circ\)[/tex]
We need to determine which equation solves for [tex]\( y \)[/tex] correctly by converting the general form into a specific one for our problem.
1. Option 1: [tex]\( 9^2 = y^2 + 19^2 - 2(y)(19) \cos 41^\circ \)[/tex]
Rewriting in terms of the Law of Cosines, it becomes:
[tex]\[ a^2 = y^2 + b^2 - 2yb \cos 41^\circ \][/tex]
(Note: This doesn't conform to solving for [tex]\( y \)[/tex].)
2. Option 2: [tex]\( y^2 = 9^2 + 19^2 - 2(y)(19) \cos 41^\circ \)[/tex]
This is not in compliance with the Law of Cosines because [tex]\( y \)[/tex] is being used inconsistently. It introduces unnecessary confusion.
3. Option 3: [tex]\( 9^2 = y^2 + 19^2 - 2(9)(19) \cos 41^\circ \)[/tex]
This equation conforms to the format:
[tex]\[ a^2 = y^2 + b^2 - 2(a)(b) \cos(A) \][/tex]
So it correctly applies the Law of Cosines.
4. Option 4: [tex]\( y^2 = 9^2 + 19^2 - 2(9)(19) \cos 41^\circ \)[/tex]
Rewriting in terms of Law of Cosines,
[tex]\[ y^2 = 9^2 + 19^2 - 2(9)(19) \cos 41^\circ \][/tex]
This correctly states the equation by solving for [tex]\( y \)[/tex] while [tex]\(a = 9\)[/tex], [tex]\(b = 19\)[/tex] respectively.
Upon verifying the options and aligning with the given angle and known sides, Option 4 is correct.
Final answer:
[tex]\[ y^2 = 9^2 + 19^2 - 2(9)(19) \cos 41^\circ \][/tex]
