Certainly! The law of cosines is a useful theorem in trigonometry which relates the lengths of the sides of a triangle to the cosine of one of its angles. It can be used to find the length of an unknown side when you are given:
1. The lengths of two sides (let's call them [tex]\( b \)[/tex] and [tex]\( c \)[/tex]).
2. The measure of the included angle (let's call it [tex]\( A \)[/tex]) between those two sides.
For the formula:
[tex]\[ a^2 = b^2 + c^2 - 2bc\cos(A) \][/tex]
We want to find the length of the unknown side [tex]\( a \)[/tex]. Let's go through the steps using specific example values:
- [tex]\( b = 5 \)[/tex]
- [tex]\( c = 7 \)[/tex]
- [tex]\( A = 50^\circ \)[/tex]
Here’s the step-by-step solution:
1. Convert the angle from degrees to radians:
The cosine function in the formula requires the angle [tex]\( A \)[/tex] to be in radians. We convert [tex]\( 50^\circ \)[/tex] to radians, knowing that [tex]\( 180^\circ = \pi \)[/tex] radians:
[tex]\[
A_{\text{radians}} = 50^\circ \times \frac{\pi}{180^\circ} = \frac{50\pi}{180} = \frac{5\pi}{18}
\][/tex]
2. Calculate the cosine of the angle:
[tex]\[
\cos(A) = \cos\left(\frac{5\pi}{18}\right)
\][/tex]
3. Substitute the values into the law of cosines formula:
[tex]\[
a^2 = b^2 + c^2 - 2bc\cos(A)
\][/tex]
Substituting [tex]\( b = 5 \)[/tex], [tex]\( c = 7 \)[/tex], and [tex]\( A = \frac{5\pi}{18} \)[/tex]:
[tex]\[
a^2 = 5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot \cos\left(\frac{5\pi}{18}\right)
\][/tex]
4. Simplify the equation:
[tex]\[
a^2 = 25 + 49 - 70 \cos\left(\frac{5\pi}{18}\right)
\][/tex]
5. Compute the numerical values:
First, calculate [tex]\( \cos\left(\frac{5\pi}{18}\right) \)[/tex]:
[tex]\[
\cos\left(\frac{5\pi}{18}\right) \approx 0.6428
\][/tex]
Substituting this cosine value back into the equation:
[tex]\[
a^2 = 25 + 49 - 70 \cdot 0.6428
\][/tex]
Compute it:
[tex]\[
a^2 = 25 + 49 - 44.996 = 29.004
\][/tex]
6. Take the square root to find [tex]\( a \)[/tex]:
[tex]\[
a = \sqrt{29.004}
\][/tex]
[tex]\[
a \approx 5.385616707670742
\][/tex]
So, the length of the unknown side [tex]\( a \)[/tex] is approximately 5.3856 units.
Remember, this method works for any triangle, provided you know two sides and the included angle or all three sides.
