Certainly! Let's break this down step-by-step to find the correct equation when solving for [tex]\(y\)[/tex] using the Law of Cosines.
The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
In this case, we are using the following values:
- One side [tex]\(a = 9\)[/tex]
- One side [tex]\(b = 19\)[/tex]
- The included angle [tex]\(C = 41^\circ\)[/tex]
- And we want to find the missing side [tex]\(c\)[/tex], which we can denote as [tex]\(y\)[/tex].
Let's go through the given equations and determine which one correctly represents the Law of Cosines for this problem.
1. [tex]\(9^2 = y^2 + 19^2 - 2(y)(19) \cos \left(41^{\circ}\right)\)[/tex]
2. [tex]\(y^2 = 9^2 + 19^2 - 2(y)(19) \cos \left(41^{\circ}\right)\)[/tex]
3. [tex]\(9^2 = y^2 + 19^2 - 2(9)(19) \cos \left(41^{\circ}\right)\)[/tex]
4. [tex]\(y^2 = 9^2 + 19^2 - 2(9)(19) \cos \left(41^{\circ}\right)\)[/tex]
Let’s analyze each equation:
- The first and third equations cannot be correct, because they erroneously place [tex]\(9\)[/tex] as both a given side and the side being solved for (denoted as [tex]\(y\)[/tex]).
- The second equation cannot be correct because it incorrectly has [tex]\(y\)[/tex] as a factor in the cosine term, which contradicts the standard form of the Law of Cosines.
- The fourth equation matches the standard form of the Law of Cosines perfectly.
Therefore, the correct equation when solving for [tex]\(y\)[/tex] is:
[tex]\[ y^2 = 9^2 + 19^2 - 2(9)(19) \cos \left(41^{\circ}\right) \][/tex]
Let's check the detailed solution:
1. Calculate [tex]\( 9^2 \)[/tex]:
[tex]\[ 9^2 = 81 \][/tex]
2. Calculate [tex]\( 19^2 \)[/tex]:
[tex]\[ 19^2 = 361 \][/tex]
3. Calculate [tex]\( 2 \cdot 9 \cdot 19 \)[/tex]:
[tex]\[ 2 \cdot 9 \cdot 19 = 342 \][/tex]
4. Calculate [tex]\( \cos(41^{\circ}) \)[/tex]:
[tex]\[ \cos(41^{\circ}) \approx 0.754709580222772 \][/tex]
5. Compute the term [tex]\( 2 \cdot 9 \cdot 19 \cdot \cos(41^{\circ}) \)[/tex]:
[tex]\[ 342 \cdot 0.754709580222772 \approx 258.11067643618804 \][/tex]
6. Substituting these values back into the equation [tex]\( y^2 = 81 + 361 - 258.11067643618804 \)[/tex]:
[tex]\[ y^2 = 81 + 361 - 258.11067643618804 \approx 183.88932356381196 \][/tex]
7. Therefore:
[tex]\[ y \approx \sqrt{183.88932356381196} \approx 13.56 \][/tex]
So, [tex]\( y^2 \approx 183.8893 \)[/tex] is consistent with the given numerics provided, ensuring our solution aligns perfectly with the scenario described.
