Certainly! The Law of Cosines can be used to determine the length of an unknown side [tex]\( a \)[/tex] of a triangle when we have one of the following sets of information:
1. We know the lengths of two sides ([tex]\(b\)[/tex] and [tex]\(c\)[/tex]) and the measure of their included angle ([tex]\(A\)[/tex]).
2. We know the lengths of all three sides (to find any angle, but that's not our goal here).
Let's consider a specific example to see this in action:
### Given:
- The length [tex]\( b = 5 \)[/tex]
- The length [tex]\( c = 7 \)[/tex]
- The angle [tex]\( A = 30^\circ \)[/tex] (which is the angle between sides [tex]\(b\)[/tex] and [tex]\(c\)[/tex])
### Step-by-Step Solution:
1. Convert the angle from degrees to radians:
Since trigonometric functions in many calculations, including cosine, typically use radians, we first convert [tex]\( 30^\circ \)[/tex] to radians.
[tex]\[
A = 30^\circ \times \left(\frac{\pi}{180^\circ}\right) = \frac{\pi}{6} \text{ radians}
\][/tex]
2. Apply the Law of Cosines:
According to the Law of Cosines,
[tex]\[
a^2 = b^2 + c^2 - 2 \cdot b \cdot c \cdot \cos(A)
\][/tex]
Substitute the known values:
[tex]\[
a^2 = 5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot \cos\left(\frac{\pi}{6}\right)
\][/tex]
3. Calculate cosine of the angle:
[tex]\[
\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}
\][/tex]
4. Plugging in the cosine value:
[tex]\[
a^2 = 25 + 49 - 2 \cdot 5 \cdot 7 \cdot \left(\frac{\sqrt{3}}{2}\right)
\][/tex]
[tex]\[
a^2 = 25 + 49 - 35\sqrt{3}
\][/tex]
5. Perform the arithmetic calculations:
[tex]\[
a^2 = 25 + 49 - 35 \cdot 0.866 = 25 + 49 - 30.31 = 13.378
\][/tex]
6. Taking the square root to find [tex]\(a\)[/tex]:
[tex]\[
a = \sqrt{13.378} \approx 3.658
\][/tex]
### Final Answer:
- The squared length of the unknown side [tex]\(a^2\)[/tex] is approximately [tex]\(13.378\)[/tex].
- The length of the unknown side [tex]\(a\)[/tex] is approximately [tex]\(3.658\)[/tex].
Therefore, by following these steps, we find that the Law of Cosines allows us to determine that the length of the unknown side [tex]\(a\)[/tex] is approximately [tex]\(3.658\)[/tex] when [tex]\( b = 5 \)[/tex], [tex]\( c = 7 \)[/tex], and [tex]\(A = 30^\circ\)[/tex].
