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# Cosine Rule|Definition & Meaning

## Definition

The cosine rule, often known as the **law of cosines**, is used to solve **triangles**. It states: “The **sum** of the squares of the lengths of any pair of **sides** of a triangle minus **two** times the product of these two sides’ lengths multiplied by the cosine of the **angle** between them equals the **square** of the side length **opposite** to this angle.”

**Figure 1** shows a triangle **PQR**. The length of the triangle’s three **sides** are **p**, **q**, and **r**, and their corresponding **angles** are **P**, **Q**, and** R**.

The **cosine** rule for the side** r** is given as:

**r ^{2} = p^{2} + q^{2} – 2pq.cos(R)**

For the same figure, the other **two** forms of **cosine** rule are:

p^{2} = q^{2} + r^{2} – 2qr.cos(P)

q^{2} = p^{2} + r^{2} – 2pr.cos(Q)

## Concept of Opposite Side

The side opposite to any given **angle** is termed the opposite side or the side opposite to that angle. Its concept is always associated with an **interior** angle of a triangle and is essential while dealing with the **cosine** rule. **Figure 2** shows a triangle with three interior angles: **L**, **M**, and **N**.

The side **opposite** to angle **L** is **l**. Similarly, the side opposite to angle **M** is **m**, and the side opposite to angle **N** is **n**.

## Solving a Triangle

A triangle is a **polygon** with three angles, three sides or **edges**, and three vertices. It is a two-dimensional closed **shape** in which no diagonal can be drawn. The **cosine** rule is used to solve the following types of triangles.

### SAS Triangle

A SAS triangle refers to a **Side-Angle-Side** triangle, which means two **adjacent** sides and the **angle** between them. The cosine rule solves a triangle in which the **factors** known are the two adjacent sides and their **corresponding** angle to find the **third** side opposite to the given angle.

### SSS Triangle

A SSS triangle refers to a **Side-Side-Side** triangle. The three sides are given, and the **three** angles are unknown. The three angles can be calculated using the three forms of the **cosine** rule to solve the triangle.

## Sum of Angles of a Triangle

The **sum** of the angles of a triangle equals **180°**. If the three **angles** of the triangle are **P**, **Q**, and **R**, their sum can be written as:

P + Q + R = 180°

It is used to calculate the **third** angle if the two **angles** of the triangle are **known**.

## Pythagoras Theorem

The **Pythagoras** theorem only applies to right-angle triangles, whereas the cosine rule applies to all triangles, so it **generalizes** the Pythagoras theorem. **Figure 3** shows a right-angle triangle **UVW**.

The **side** opposite to the perpendicular (**90°)** angle is the longest side, known as the **hypotenuse** **h**. The base **b** is the side **adjacent** to the right angle, and the perpendicular** p** is the **height** of the triangle. The Pythagoras theorem is given as:

**h ^{2} = p^{2} + b^{2}**

The **cosine** rule is given as:

r^{2} = p^{2} + q^{2} – 2pq.cos(R)

If the angle **R** is taken as **90°**, then the triangle is a **right-angle** triangle. Putting the value in the **cosine** rule gives:

r^{2} = p^{2} + q^{2} – 2pq.cos(90°)

As:

cos(90°) = 0

So:

**r ^{2} = p^{2} + q^{2}**

Which is the same as the **Pythagorean** theorem.

## Derivation of Cosine Rule

**Figure 4** shows a triangle **PQR**. A perpendicular **PM** is drawn from vertex **P** to meet the side **QR** at point **M**.

In triangle** PQM:**

cos Q = QM / r

**QM = r.cos Q**

Similarly, in triangle **PMR:**

cos R = RM / q

**RM = q.cos R**

In triangle **PMR**, the **Pythagoras** theorem is given as:

PR^{2} = PM^{2} + RM^{2}

Also, **RM** can be written as:

PR^{2} = PM^{2} + (QR – QM)^{2}

Expanding the above equation gives:

PR^{2} = PM^{2} + QR^{2} + QM^{2} – 2.QR.QM

Also:

**PM ^{2} + QM^{2} = PQ^{2}**

Replacing (PM^{2} + QM^{2}) with PQ^{2} in the above equation:

PR^{2} = PQ^{2} + QR^{2} – 2.QR.QM

As we know from **figure 4:**

QR = p, PR = q, PQ = r, QM = r.cos Q

Replacing **PR**, **PQ**, **QR**, and **QM** in the above equation gives:

q^{2} = r^{2} + p^{2} – 2.p.(r.cos Q)

which is the cosine rule and is the same as:

**q ^{2} = p^{2} + r^{2} – 2pr.cos Q**

**Rearranging** the above equation gives:

cos Q = (p^{2} + r^{2} – q^{2}) / 2pr

Similarly, we can also prove the other **forms** of the cosine rule.

## Examples

### Example 1 – SAS Triangle

A triangle PQR is shown in **figure 5**. Find the length of the **third** side using the cosine rule.

#### Solution

The **length** of two adjacent sides, **p** and **r** are given as **5** and **9**, respectively. The angle between sides **p** and r is **62.2°**. The **cosine** rule for side **q** is given as:

q^{2} = p^{2} + r^{2} – 2pr.cos(Q)

Here,

p = 5, r = 9, Q = 62.2°

Putting the **values** in the above equation gives:

q^{2} = (5)^{2} + (9)^{2} – 2(5)(9).cos(62.2°)

q^{2} = 25 + 81 – 42

q^{2} = 64

**q = 8**

Hence, the length of the **side** opposite to angle **Q** is **8**.

### Example 2 – SSS Triangle

**Figure 6** shows a triangle **PQR**. Find the three **angles** of the triangle.

#### Solution

The **length** of the triangle’s three sides, **p**, **q**, and **r**, is given as **11**, **8**, and **6**. Using the cosine rule for angle **P** as:

p^{2} = q^{2} + r^{2} – 2qr.cos(P)

**Rearranging** the above equation gives:

cos(P) = (q^{2} + r^{2} – p^{2}) / 2qr

Here:

p = 11, q = 8, r = 6

Putting the values gives:

cos(P) = [(8)^{2} + (6)^{2} – (11)^{2}] / 2(8)(6)

cos(P) = [64 + 36 – 121] / 96

cos(P) = – 21 / 96

cos(P) = – 0.21875

**P = 102.6°**

Using the **cosine** rule for angle **Q** as:

cos Q = (p^{2} + r^{2} – q^{2}) / 2pr

Putting the values of **p**, **q**, and **r** gives:

cos Q = [(11)^{2} + (6)^{2} – (8)^{2}] / 2(11)(6)

cos Q = [121 + 36 – 64] / 2(11)(6)

cos Q = [121 + 36 – 64] / 132

cos Q = 93 / 132

cos Q = 0.7045

**Q = 45.2°**

As the **sum** of three angles of a triangle is **180°**, so:

P + Q + R = 180**°**

For the angle **R**, rearranging the equation gives:

R = 180**°** – P – Q

Putting the values of **P** and **Q** gives:

R = 180**°** – 102.6**°** – 45.2**°**

**R = 32.2°**

*All the images are created using Geogebra.*