Certainly! Let's delve into the problem and solve it step-by-step using the Law of Cosines.
### Problem Statement:
You need to find the length of side [tex]\( a \)[/tex] in a triangle given the lengths of sides [tex]\( b \)[/tex] and [tex]\( c \)[/tex], and the angle [tex]\( A \)[/tex] between them.
The Law of Cosines states:
[tex]\[
a^2 = b^2 + c^2 - 2bc \cos(A)
\][/tex]
where:
- [tex]\( a \)[/tex] is the side opposite angle [tex]\( A \)[/tex]
- [tex]\( b \)[/tex] and [tex]\( c \)[/tex] are the other two sides
- [tex]\( A \)[/tex] is the included angle between sides [tex]\( b \)[/tex] and [tex]\( c \)[/tex]
### Given Values:
- [tex]\( b = 0 \)[/tex]
- [tex]\( c = 0 \)[/tex]
- [tex]\( A = 0 \)[/tex] radians
### Steps to Solve for [tex]\( a \)[/tex]:
1. Substitute the given values into the Law of Cosines formula:
[tex]\[
a^2 = b^2 + c^2 - 2 \cdot b \cdot c \cdot \cos(A)
\][/tex]
Plugging in the values, we get:
[tex]\[
a^2 = 0^2 + 0^2 - 2 \cdot 0 \cdot 0 \cdot \cos(0)
\][/tex]
2. Simplify each term:
- [tex]\( 0^2 = 0 \)[/tex]
- Cosine of 0 radians, [tex]\(\cos(0)\)[/tex], is 1.
Therefore, the equation simplifies to:
[tex]\[
a^2 = 0 + 0 - 2 \cdot 0 \cdot 0 \cdot 1
\][/tex]
[tex]\[
a^2 = 0 - 0
\][/tex]
[tex]\[
a^2 = 0
\][/tex]
3. Take the square root of both sides to solve for [tex]\( a \)[/tex]:
[tex]\[
a = \sqrt{0}
\][/tex]
[tex]\[
a = 0
\][/tex]
### Conclusion:
The length of side [tex]\( a \)[/tex] is [tex]\( 0 \)[/tex].
In summary, after applying the Law of Cosines and substituting the given values, we find that the length of side [tex]\( a \)[/tex] is [tex]\( 0.0 \)[/tex].
