Trigonometry: Law of Sines

The Law of Sines, also known as the Sine Rule or Sine Law, is a fundamental principle in trigonometry used to solve triangles. It relates the sides of a triangle to the sines of its angles. The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. This law is expressed mathematically as follows:

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

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

This law is particularly useful when dealing with non-right triangles, where the familiar relationships of the Pythagorean Theorem and trigonometric functions involving right angles do not apply directly.

Derivation of the Law of Sines

To understand why the Law of Sines holds true, we can derive it using the unit circle and basic trigonometric relationships. Consider a triangle with sides $a, b,$ and $c$, and angles $A, B,$ and $C$.

Let’s focus on angle $A$ and its opposite side $a$. Draw a perpendicular from vertex $A$ to side $a$, creating two right triangles within the larger triangle. Label the length of the perpendicular as $h$.

Using the definition of the sine function, we have:

$\sin A = \frac{h}{c}$

This is because the sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the hypotenuse. Rearranging the equation gives:

$h = c \cdot \sin A$

Now, consider the right triangle formed by side $b$, $h$, and the segment of side $c$ adjacent to angle $A$. By the definition of sine again, we have:

$\sin C = \frac{h}{b}$

Rearranging this equation gives:

$h = b \cdot \sin C$

Since both expressions equal $h$, we can set them equal to each other:

$c \cdot \sin A = b \cdot \sin C$

Dividing both sides by $c$ gives:

$\frac{\sin A}{c} = \frac{\sin C}{b}$

Multiplying both sides by $a$ (the side opposite angle $A$) results in:

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

Similarly, this process can be repeated for angles $B$ and $C$, yielding the other equal ratios in the Law of Sines.

Applications of the Law of Sines

1. Solving Triangles: The Law of Sines is commonly used to solve triangles when given certain information. For example, if you know two angles and one side (not the one opposite an angle), you can use the Law of Sines to find the other sides and angles.

2. Navigation: In navigation and map reading, the Law of Sines can be used to determine distances or angles between points, especially when dealing with oblique triangles.

3. Astronomy: Astronomers use the Law of Sines to calculate distances in space, such as determining the distance to a star based on observations from different points on Earth.

4. Engineering and Physics: The Law of Sines is also applied in various engineering and physics problems involving triangular structures or forces acting at angles.

Conditions for Using the Law of Sines

It’s important to note that the Law of Sines applies to any triangle, not just right triangles. However, there are certain conditions and limitations to keep in mind when using this law:

1. Ambiguity: The Law of Sines can lead to an ambiguous case when solving triangles. This occurs when, given certain information, two different triangles can be constructed. In such cases, additional information or a different approach may be needed to determine the correct triangle.

2. Side-Angle-Side (SAS): The Law of Sines is not directly applicable when you know two sides and the angle opposite one of them (SAS case). In this scenario, you would typically use the Law of Cosines or other triangle-solving techniques.

3. Angle-Side-Angle (ASA): If you know two angles and the side between them (ASA case), the Law of Sines can be used to find the remaining side.

4. Angle-Angle-Side (AAS): When you have two angles and a side not between them (AAS case), the Law of Sines can be applied to find the missing sides or angles.

Example Problem Using the Law of Sines

Let’s work through an example problem to demonstrate how the Law of Sines is applied:

Problem: In triangle $ABC$, angle $A$ measures $40^\circ$, angle $B$ measures $75^\circ$, and side $a$ opposite angle $A$ has a length of 8 units. Find the lengths of sides $b$ and $c$.

Solution: We are given angle $A$, angle $B$, and side $a$. We can use the Law of Sines to find the lengths of sides $b$ and $c$.

First, set up the proportion based on the Law of Sines:

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

Plugging in the given values:

$\frac{8}{\sin 40^\circ} = \frac{b}{\sin 75^\circ} = \frac{c}{\sin C}$

Now, solve for $b$ and $c$ separately:

$b = 8 \cdot \frac{\sin 75^\circ}{\sin 40^\circ} \approx 11.78 \, \text{units}$

$c = 8 \cdot \frac{\sin C}{\sin 40^\circ}$

We need to find $\sin C$. Since the sum of angles in a triangle is $180^\circ$, we can find angle $C$:

$C = 180^\circ – A – B = 180^\circ – 40^\circ – 75^\circ = 65^\circ$

$\sin C = \sin 65^\circ$

Plugging in the values:

$c = 8 \cdot \frac{\sin 65^\circ}{\sin 40^\circ} \approx 12.98 \, \text{units}$

Therefore, the lengths of sides $b$ and $c$ are approximately 11.78 units and 12.98 units, respectively.

Conclusion

The Law of Sines is a powerful tool in trigonometry, enabling us to solve triangles and calculate unknown sides or angles. Its application extends to various fields, including mathematics, physics, engineering, and navigation. Understanding how to apply this law correctly and being aware of its limitations are essential for solving trigonometric problems effectively.