To find the length of side \( c \) in a triangle where sides \( a = 5 \), \( b = 1 \), and angle \( C = 40^\circ \), we can use the Law of Cosines. The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
Here is the step-by-step solution:
1. Convert the angle from degrees to radians:
Since the formula uses the cosine of the angle, and most calculators or computations work with radians, we convert \( 40^\circ \) to radians:
[tex]\[
C = 40^\circ = 0.6981 \text{ radians} \quad \text{(approx)}
\][/tex]
2. Apply the Law of Cosines:
Substitute the values into the Law of Cosines formula:
[tex]\[
c^2 = 5^2 + 1^2 - 2 \cdot 5 \cdot 1 \cdot \cos(40^\circ)
\][/tex]
[tex]\[
c^2 = 25 + 1 - 10 \cdot \cos(0.6981)
\][/tex]
3. Calculate the cosine value:
The cosine of \( 0.6981 \) radians is approximately \( 0.766 \) (rounded to three decimal places).
4. Complete the formula:
[tex]\[
c^2 = 26 - 10 \cdot 0.766
\][/tex]
[tex]\[
c^2 = 26 - 7.66
\][/tex]
[tex]\[
c^2 = 18.34 \quad \text{(rounded to two decimal places)}
\][/tex]
5. Solve for \( c \):
Take the square root of both sides to find \( c \):
[tex]\[
c = \sqrt{18.34} \approx 4.282 \quad \text{(rounded to three decimal places)}
\][/tex]
Hence, the length of side [tex]\( c \)[/tex] is [tex]\( 4.282 \)[/tex].
