Law of Sine and Cosine Rule

Introduction

• Laws of Sine and Cosine are trigonometric formulas used to solve triangles when certain information is given.
• They are especially useful for non-right triangles.
• These laws are fundamental in trigonometry and have applications in physics, engineering, and navigation.

What is the Sine Rule?

• The Sine Rule is a fundamental trigonometric formula that relates the sides of a triangle to the sines of their opposite angles.
• Mathematically,

• $a$, $b$ and $c$ are the lengths of the sides of the triangle.
• $A$, $B$ and $C$ are the angles opposite those sides.

Solved Example

Given $A = 40^\circ$, $B = 60^\circ$ and side $a = 10$ cm. Find side $b$.

SOLUTION

Use the formula:

$$\frac{b}{\sin B} = \frac{a}{\sin A}$$

Put the values:

$$\frac{b}{\sin 60^\circ} = \frac{10}{\sin 40^\circ}$$

Now calculate using a calculator:

$$b = \frac{10 \cdot \sin 60^\circ}{\sin 40^\circ}$$

Final Answer: $b = 13.5$ cm

What is the Cosine Rule?

• The Cosine Rule is also a trigonometric formula used to find a side or angle in a triangle.
• It works for any triangle whether it’s acute, obtuse, or right-angled.
• Mathematically,

• $a$, $b$ and $c$ are the lengths of the sides of the triangle.
• $A$, $B$ and $C$ are the angles opposite those sides.

Solved Example

Side $a = 5$ cm, side $b = 7$ cm, angle $C = 60^\circ$. Find side $c$.

SOLUTION

Use the formula:

$$c^2 = a^2 + b^2 – 2ab \cos C$$

Put the values:

$$c^2 = 5^2 + 7^2 – 2(5)(7) \cos 60^\circ$$

$$c^2 = 25 + 49 – 70(0.5) = 39$$

$$c = \sqrt{39} \approx 6.24$$

Final Answer: $c = 6.24$ cm

How to Find Missing Side and Angle?

• The Sine Rule or the Cosine Rule, both are used to find the missing side or missing angle depending on what information is given in the question.

• If we know the 2 angles and one side, then we use it to find another side.

• If we know the 2 sides and one non-included angle, then we use it to find the other angle.

• If we know the 2 sides and one included angle, then we use it to find third side.

• If we know all the three sides, then we use it to find any angle.

Solved Examples

Solved Example

In Triangle $ABC$, Side $a = 10$ cm, Side $b = 14$ cm and Angle $A = 45^\circ$. Find angle $B$.

SOLUTION

1

Given:

• Side $a = 10$ cm
• Side $b = 14$ cm
• Angle $A = 45^\circ$

2

Use The Formula:

$$\frac{\sin B}{b} = \frac{\sin A}{a}$$

3

Plug the values:

$$\frac{\sin B}{14} = \frac{\sin 45^\circ}{10}$$

4

Solve for the missing angle:

$$\sin B = \frac{14 \cdot \sin 45^\circ}{10}$$

The missing angle of $B \approx 81.6^\circ$

Final Answer: $B \approx 81.6^\circ$

Solved Example

In Triangle $ABC$, Side $a = 7$ cm, Side $b = 8$ cm and Side $c = 9$ cm. Find angle $C$.

SOLUTION

1

Given:

• Side $a = 7$ cm
• Side $b = 8$ cm
• Side $c = 9$ cm

2

Use The Formula:

$$\cos C = \frac{a^2 + b^2 – c^2}{2ab}$$

3

Plug the values:

$$\cos C = \frac{7^2 + 8^2 – 9^2}{2(7)(8)}$$

4

Solve for the missing angle:

$$\cos C = \frac{49 + 64 – 81}{112} = \frac{32}{112}$$

The missing angle of $C \approx 73.4^\circ$

Final Answer: $C \approx 73.4^\circ$

Solved Example

In Triangle $ABC$, Angle $A = 50^\circ$, Angle $B = 60^\circ$ and Side $a = 10$ cm. Find side $b$.

SOLUTION

1

Given:

• Side $a = 10$ cm
• Angle $A = 50^\circ$
• Angle $B = 60^\circ$

2

Use The Formula:

$$\frac{b}{\sin B} = \frac{a}{\sin A}$$

3

Plug the values:

$$\frac{b}{\sin 60^\circ} = \frac{10}{\sin 50^\circ}$$

4

Solve for the missing Side:

$$b = \frac{10 \cdot \sin 60^\circ}{\sin 50^\circ}$$

The missing side of $b \approx 11.31$ cm

Final Answer: $b \approx 11.31$ cm