To find the length of the third side of a triangle where we know two sides and the included angle, we can use the Law of Cosines. The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \][/tex]
where:
- [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the two known sides,
- [tex]\( C \)[/tex] is the included angle, and
- [tex]\( c \)[/tex] is the length of the third side we are trying to find.
Given:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 5 \)[/tex]
- [tex]\( C = 60^{\circ} \)[/tex]
First, we need to convert the angle from degrees to radians, because the cosine function typically expects an angle in radians. The conversion formula is:
[tex]\[ \text{angle in radians} = \text{angle in degrees} \times \left( \frac{\pi}{180} \right) \][/tex]
So:
[tex]\[ 60^{\circ} \times \left( \frac{\pi}{180} \right) = \frac{\pi}{3} \approx 1.047 \text{ radians} \][/tex]
Next, we substitute the given values into the cosine rule formula:
[tex]\[ c^2 = 2^2 + 5^2 - 2 \cdot 2 \cdot 5 \cdot \cos(60^{\circ}) \][/tex]
We know that [tex]\( \cos(60^{\circ}) = 0.5 \)[/tex], so:
[tex]\[ c^2 = 4 + 25 - 2 \cdot 2 \cdot 5 \cdot 0.5 \][/tex]
[tex]\[ c^2 = 4 + 25 - 2 \cdot 2 \cdot 5 \cdot 0.5 \][/tex]
[tex]\[ c^2 = 4 + 25 - 10 \][/tex]
[tex]\[ c^2 = 19 \][/tex]
To find [tex]\( c \)[/tex], we take the square root of both sides:
[tex]\[ c = \sqrt{19} \][/tex]
Using a calculator, we find:
[tex]\[ c \approx 4.359 \][/tex]
Therefore, the length of the third side of the triangle is approximately [tex]\( 4.359 \)[/tex].
