Answer :
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.
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.
