6.2 Law of Cosines
Law of Cosines
a² = b² + c² - 2bc cos A cos A = (b² + c² -a²)/(2bc)
b² = a² + c² - 2ac cos B cos B = (a² + c² -b²)/(2ac)
c² = a² + b² - 2ab cos C cos C = (a² + b² - c²)/(2ab)
a² = b² + c² - 2bc cos A cos A = (b² + c² -a²)/(2bc)
b² = a² + c² - 2ac cos B cos B = (a² + c² -b²)/(2ac)
c² = a² + b² - 2ab cos C cos C = (a² + b² - c²)/(2ab)
Three Sides of a Triangle SSS
The largest angle should be found first. Knowing the cosine of an angle, you can determine whether the angle is acute or obtuse.
cos θ > 0 for 0° < θ < 90°
cos θ < 0 for 90° < θ <180°
So after finding the largest angle, the next two angles must be acute.
cos θ > 0 for 0° < θ < 90°
cos θ < 0 for 90° < θ <180°
So after finding the largest angle, the next two angles must be acute.
Two Sides and the Included Angle SAS
Applications
Heron's Area Formula
The Law of Cosines can be used to establish the following formula for the area of a triangle, called Heron's Area Formula.
Given any triangle with sides of lengths a, b, and c, the area of the triangle is given by
Area = √(s(s - a)(s - b)(s - c))
where s = (a + b + c)/2
The Law of Cosines can be used to establish the following formula for the area of a triangle, called Heron's Area Formula.
Given any triangle with sides of lengths a, b, and c, the area of the triangle is given by
Area = √(s(s - a)(s - b)(s - c))
where s = (a + b + c)/2
Using Heron's Area Formula