To solve for [tex]\( c \)[/tex] given the equation [tex]\( c^2 = a^2 + b^2 - 2ab \cos C \)[/tex], where [tex]\( a = 3 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( C = 60^\circ \)[/tex], follow these steps:
1. Complete the equation with known values:
[tex]\[
c^2 = 3^2 + 3^2 - 2 \cdot 3 \cdot 3 \cdot \cos(60^\circ)
\][/tex]
2. Calculate the values inside the equation:
[tex]\[
3^2 = 9
\][/tex]
Now substitute these back into the equation:
[tex]\[
c^2 = 9 + 9 - 2 \cdot 3 \cdot 3 \cdot \cos(60^\circ)
\][/tex]
3. Calculate the cosine of [tex]\( 60^\circ \)[/tex]:
[tex]\[
\cos(60^\circ) = \frac{1}{2}
\][/tex]
Substitute this back into the equation:
[tex]\[
c^2 = 9 + 9 - 2 \cdot 3 \cdot 3 \cdot \frac{1}{2}
\][/tex]
4. Simplify the multiplication:
[tex]\[
2 \cdot 3 \cdot 3 \cdot \frac{1}{2} = 9
\][/tex]
Substitute this into the equation:
[tex]\[
c^2 = 9 + 9 - 9
\][/tex]
5. Combine like terms:
[tex]\[
c^2 = 9
\][/tex]
6. Find [tex]\( c \)[/tex] by taking the square root of both sides:
[tex]\[
c = \sqrt{9} = 3
\][/tex]
Therefore, [tex]\( c \)[/tex] is [tex]\( 3 \)[/tex].
