Trigonometry: Law of Sines

The law of sines, also known as the sine rule or sine law, is a fundamental concept in trigonometry that relates the sides of a triangle to the sines of its angles. It 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 particularly useful in solving triangles when you have information about the lengths of sides and/or the measures of angles.

The law of sines can be 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.
• $A$, $B$, and $C$ are the measures of the angles opposite to sides $a$, $b$, and $c$ respectively.

This law essentially states that the ratio of the length of a side to the sine of the angle opposite that side is the same for all three sides of the triangle. This is true for any triangle, whether it is acute, obtuse, or right-angled.

The law of sines can be derived from the properties of similar triangles. Consider a triangle ABC with sides of lengths $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. Let’s drop an altitude from vertex $A$ to side $BC$, creating two right triangles: one with side lengths $h$ (the altitude), $b’$ (the segment of $BC$ from $A$ to the foot of the altitude), and angle $A$, and another with side lengths $h$, $c’$ (the segment of $BC$ from the foot of the altitude to $C$), and angle $C$.

Using the definition of the sine function, we have:

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

Rearranging these equations gives us:

$h = b’ \sin A$
$h = c’ \sin C$

Since the altitude $h$ is the same for both right triangles (as they share the same base $BC$), we can set these two expressions for $h$ equal to each other:

$b’ \sin A = c’ \sin C$

Dividing both sides by $b’$ and $c’$ respectively gives:

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

Since angles $A$ and $C$ are not necessarily equal (unless the triangle is isosceles or equilateral), we can express this relationship as:

$\sin A = \sin (180^\circ – C)$

Using the fact that $\sin (180^\circ – \theta) = \sin \theta$, we simplify to:

$\sin A = \sin C$

This establishes the relationship between the angles and their opposite sides in any triangle. Extending this reasoning to the other pairs of angles and sides in the triangle, we arrive at the general form of the law of sines:

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

The law of sines is especially useful in solving triangles when you have information about angles and sides but not enough information to apply the law of cosines or other methods. It allows you to find missing side lengths or angles in a triangle when you know the measurements of some angles and/or sides. However, it’s important to note that the law of sines may not be applicable or may yield ambiguous results in certain cases, such as when trying to solve triangles with inadequate information or when dealing with obtuse-angled triangles.

Certainly! Let’s delve deeper into the law of sines and explore some additional aspects and applications of this fundamental trigonometric concept.

Derivation from the Unit Circle:

The law of sines can also be derived geometrically from the unit circle. Consider a unit circle centered at the origin of a Cartesian coordinate system. Draw a radius from the origin to a point on the circle, forming an angle $\theta$ with the positive x-axis. The coordinates of this point are $(\cos \theta, \sin \theta)$.

Now, draw a perpendicular line from this point to the x-axis, creating a right triangle. The length of the side opposite angle $\theta$ is $\sin \theta$, and the length of the side adjacent to angle $\theta$ is $\cos \theta$.

If we scale this triangle by a factor of $r$, where $r$ is the radius of the circle (the hypotenuse of the triangle), the side opposite angle $\theta$ becomes $r \sin \theta$. This is a key insight for understanding the law of sines.

Ambiguous Case:

One important consideration when using the law of sines is the ambiguous case, also known as the “SSA” case, which stands for “side-side-angle.” In this case, if you are given two sides of a triangle and the angle opposite one of those sides, there may be two possible triangles that satisfy these conditions, one with an acute angle and one with an obtuse angle. This ambiguity arises because the sine function is periodic.

To determine whether the ambiguous case exists and to find the correct triangle(s), you may need additional information such as the length of another side or the measures of additional angles.

Applications:

1. Navigation and Surveying: The law of sines is used in navigation and surveying to determine distances and angles. For example, sailors use trigonometry to navigate using the positions of celestial bodies, and surveyors use it to measure distances and create accurate maps.

2. Astronomy: Astronomers use trigonometry extensively to study celestial objects and phenomena. The law of sines helps calculate distances to stars, planets, and other celestial bodies.

3. Engineering and Architecture: Trigonometry is essential in engineering and architecture for designing structures, calculating forces, and determining angles and distances.

4. Physics: Trigonometry plays a crucial role in physics, particularly in fields like mechanics, optics, and waves. The law of sines is used in analyzing wave behavior, light refraction, and more.

5. Computer Graphics: Trigonometry is fundamental in computer graphics for rendering 3D scenes, animations, and simulations. Algorithms for rotating objects, calculating perspectives, and determining object positions often rely on trigonometric principles including the law of sines.

Extensions and Variations:

1. Law of Cosines: The law of cosines is another fundamental trigonometric law that relates the lengths of a triangle’s sides to the cosine of one of its angles. It is particularly useful for solving triangles when you know the lengths of all three sides but not the angles.

2. Spherical Trigonometry: In spherical trigonometry, which deals with triangles on the surface of a sphere, there are analogous laws to the law of sines and the law of cosines that take into account the curvature of the sphere.

3. Inverse Trigonometric Functions: The inverse trigonometric functions (such as arcsine, arccosine, and arctangent) are used to find angles given side lengths or to solve trigonometric equations.

4. Trigonometric Identities: Trigonometric identities, such as the Pythagorean identities and the sum and difference identities, are essential for simplifying trigonometric expressions and solving trigonometric equations.

Practical Examples:

Let’s consider a practical example to illustrate the application of the law of sines:

Example: In triangle $ABC$, angle $A = 30^\circ$, side $a = 8$ units, and side $b = 10$ units. Find the length of side $c$ and the measures of angles $B$ and $C$.

Using the law of sines:

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

Plugging in the given values:

$\frac{8}{\sin 30^\circ} = \frac{10}{\sin B}$

Solving for $\sin B$, we get:

$\sin B = \frac{10}{8} \times \sin 30^\circ$
$\sin B = \frac{5}{4} \times \frac{1}{2}$
$\sin B = \frac{5}{8}$

Using the inverse sine function to find angle $B$:

$B = \arcsin \left( \frac{5}{8} \right)$
$B \approx 38.66^\circ$

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

$C = 180^\circ – A – B$
$C = 180^\circ – 30^\circ – 38.66^\circ$
$C \approx 111.34^\circ$

Finally, using the law of sines again to find side $c$:

$\frac{a}{\sin A} = \frac{c}{\sin C}$
$\frac{8}{\sin 30^\circ} = \frac{c}{\sin 111.34^\circ}$

Solving for $c$, we get:

$c = \frac{8 \times \sin 111.34^\circ}{\sin 30^\circ}$
$c \approx 15.45 \text{ units}$

So, in this example, side $c$ is approximately 15.45 units, angle $B$ is approximately $38.66^\circ$, and angle $C$ is approximately $111.34^\circ$.

This example demonstrates how the law of sines can be used to solve for unknown side lengths and angles in a triangle when certain information is given.