Table of Contents

## Cosine Rule

The cosine rule is a trigonometric equation that is used to find the length of a side of a triangle when two other sides and the angle between them are known. The equation is:

cos(A) = adjacent/hypotenuse

Where A is the angle between the two known sides, adjacent is the length of the shorter side, and hypotenuse is the length of the longer side.

### Law of Cosines Problems and Solutions

The law of cosines can be used to solve problems that involve the three sides and angles of a triangle. The law states that the sum of the squares of the lengths of the two sides that are not the hypotenuse is equal to the square of the length of the hypotenuse plus twice the product of the lengths of the two sides that are not the hypotenuse.

The law of cosines can be used to solve problems that involve the three sides and angles of a triangle. The law states that the sum of the squares of the lengths of the two sides that are not the hypotenuse is equal to the square of the length of the hypotenuse plus twice the product of the lengths of the two sides that are not the hypotenuse.

The law of cosines can be used to solve problems that involve the three sides and angles of a triangle. The law states that the sum of the squares of the lengths of the two sides that are not the hypotenuse is equal to the square of the length of the hypotenuse plus twice the product of the lengths of the two sides that are not the hypotenuse.

The law of cosines can be used to solve problems that involve the three sides and angles of a triangle. The law states that the sum of the squares of the lengths of the two sides that are not the hypotenuse is equal to the square of the length of the hypotenuse plus twice the

### Proof of Law of Cosine Equation

The proof of the law of cosine equation is as follows:

Let A and B be two points on a unit circle, with coordinates (x, y) and (x+h, y+k), respectively.

Then, the cosine of the angle between A and B is given by:

cos(angle) = (x+h)^2+(y+k)^2/ (x-h)^2+(y-k)^2

Since the points A and B are on a unit circle, their coordinates are all positive.

Hence, the denominator in the equation above is always positive, and so the cosine of the angle between A and B is always positive.

### 1. To Find All the Angles of a Triangle Whose All Three Sides are Known:

The angles of a triangle can be found by using the trigonometric functions sine, cosine, and tangent. In order to find the angles of a triangle, you must first know the length of all three sides of the triangle.

Once you have the lengths of all three sides, you can use the following equation to find the angles of the triangle:

angle A = sin-1(B/2A)

angle B = sin-1(C/2B)

angle C = sin-1(A/2C)

### 2. To Find the Lengths of the Sides of the Triangle When Two Other the Sides and the Angle Between them are Known:

The length of the side opposite the given angle is the length of the hypotenuse. The other two sides are the length of the other two angles.

### Law of Cosines SSS

The law of cosines is a theorem in Euclidean geometry that states that the sum of the squares of the sides of a triangle is equal to the square of the length of the triangle’s hypotenuse.