use the law of cosines to solve the triangle A=55 degree, c=10, b=3
a^2 = 10^2 + 3^2 - 2(10)(3)cos55°
= 109 - 60cos55
= 107.8528..
a = 10.3852..
To solve the triangle using the Law of Cosines, follow these steps:
1. Identify the given values:
- Angle A is 55 degrees: A = 55°
- Side c is 10 units long: c = 10
- Side b is 3 units long: b = 3
2. Determine the missing side length using the Law of Cosines formula, which is:
c² = a² + b² - 2ab * cos(C)
In this case, we need to find side a. The corresponding angle C can be calculated by subtracting angles A and B from 180 degrees: C = 180° - A - B
Since angle B is not given, we can use the fact that the sum of angles in a triangle is 180 degrees to find it: B = 180° - A - C
Similarly, we also need to determine the value of angle C.
3. Calculate angle C:
C = 180° - A - B
C = 180° - 55° - B
C = 125° - B
4. Calculate angle B:
Since the sum of angles in a triangle is 180 degrees, we can substitute the values of angles A and C into this equation and solve for B:
A + B + C = 180°
55° + B + 125° - B = 180°
180° = 180°
Since B cancels out on both sides, B does not depend on the given values and is not needed for solving the triangle.
5. Now we can use the Law of Cosines to find the missing side length, a:
c² = a² + b² - 2ab * cos(C)
10² = a² + 3² - 2 * a * 3 * cos(125°)
100 = a² + 9 - 6a * cos(125°)
6. Rearrange the equation and solve for a:
Subtract 9 from both sides:
a² - 6a * cos(125°) = 91
Use the cosine of 125 degrees:
a² - 6a * [-0.5736] = 91
Simplify:
a² + 3.442a - 91 = 0
7. Solve the quadratic equation:
This equation can be factored or solved using the quadratic formula. Factoring is not possible in this case, so we'll use the quadratic formula:
a = (-b ± √(b² - 4ac)) / 2a
The values needed are:
a = 1
b = 3.442
c = -91
Substituting these into the formula:
a = (-3.442 ± √(3.442² - 4 * 1 * -91)) / (2 * 1)
Simplify:
a = (-3.442 ± √(11.835 + 364)) / 2
a = (-3.442 ± √375.835) / 2
a ≈ (-3.442 ± 19.384) / 2
The two possible solutions for a are:
a₁ ≈ (-3.442 + 19.384) / 2
≈ 15.942 / 2
≈ 7.971
or
a₂ ≈ (-3.442 - 19.384) / 2
≈ -22.826 / 2
≈ -11.413
Since we are dealing with side lengths, a negative value is not meaningful. Therefore, the length of side a is approximately 7.971 units.
So, the lengths of the sides of the triangle are:
a ≈ 7.971
b = 3
c = 10