Law of Cosines Formula -

The Law of Cosines is a trigonometric formula used to find the length of a side or the measure of an angle in a triangle when you have enough information about the other sides and angles. It applies to any type of triangle, whether it is right-angled or not.

The formula for the Law of Cosines is as follows:

$c^2 = a^2 + b^2 - 2ab * cos(C)$

In this formula:

‘c’ represents the length of the side opposite to angle C.
‘a’ and ‘b’ represent the lengths of the other two sides of the triangle.
‘C’ represents the angle between sides ‘a’ and ‘b’.

The Law of Cosines states that the square of the length of one side of a triangle is equal to the sum of the squares of the other two sides, minus twice the product of the lengths of those two sides, multiplied by the cosine of the included angle.

Here’s an example to illustrate the Law of Cosines:

Consider a triangle with side lengths a = 5, b = 7, and angle C = 40 degrees. We can use the Law of Cosines to find the length of side c.

$c^2 = a^2 + b^2 - 2ab * cos(C)$
$c^2 = 5^2 + 7^2 - 2(5)(7) * cos(40°)$

Simplifying the equation further:

$c^2 = 25 + 49 - 70cos(40°)$

Finally, we can find ‘c’ by taking the square root of both sides of the equation:

$c = √(25 + 49 - 70cos(40°))$

Here is a diagram to help visualize the triangle in the Law of Cosines example:

      /|   
   a / | c
    /  |
   /   |
 A/____|\
    b   C

In the diagram, ‘a’, ‘b’, and ‘c’ represent the lengths of the sides of the triangle, while ‘A’ and ‘C’ represent the angles, with angle ‘C’ being the included angle between sides ‘a’ and ‘b’.

Law of Cosines Formula

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