28 November 2023

Mathematics is an extensive area of ​​study and has an endless number of terms and tools that can be used by many other sciences. Thus, Mathematics has various branches that are dedicated to the study of subtopics that start in it, such as trigonometry.

For his part, trigonometry deals with the study and measurement of triangles, the relationships between its sides and angles and its trigonometric functions, such as sine, cosine, tangent, cotangent, etc. On this occasion, we will talk a little about the theorem of sines, what it is about, how it is used and what its mathematical formula is.

## What is the law of sines about?

The law of sines is a very important concept in trigonometry, which establishes that there is a relationship between the angles and lengths of the sides of a triangle. This defines us that the ratio between the length of a side of a triangle and the sine of the angle opposite that side will be the same for all sides of the triangle.

This theorem is considered fundamental and very useful when solving problems of trigonometry and geometry. This is thanks to the fact that it allows calculating the measure of a segment, be it a side or an angle of a triangle, as long as the measures of the other two sides and their opposite angle are known.

In addition, the law of sines can also be essential when performing other operations that involve calculating the area of ​​geometric figures, in this case, triangles, whether they are right triangles, oblique triangles, among others.

## What is the formula of the law of sines?

The law of sines defines the relationship between the sides of a triangle and its angles. This states that the length of one side of the triangle, divided by the sine of the angle opposite that side, will be equal to the length of any other side divided by the sine of the angle opposite that side.

Although it may sound a bit complicated, the relationship can be expressed in a mathematical formula as follows: a/sin(A) = b/sin(B) = c/sin(C).

where the values a, b and c are the lengths of the sides of a triangle, while A, B, and C the opposite angles of each of those sides.

## When should the law of sine and cosine be used?

Both the sine and cosine theorems They are important problem-solving tools.but they are used in different situations.

The law of sines It is usually used to find the measure of an angle or side of a triangle, knowing the measures of the other two sides and the opposite angle of one of them. Like, for example, if we know the measures of sides “a” and “b” of a triangle, and the angle “A” (opposite “a”), we can use the law of sines to find side “c” or angle “C”.

In this example, the theorem of sines would be applied, clearing the measure that we want to find, if we want to find the side “c” the formula would be: c = (a * sin(C)) / sin(A). On the other hand, if you want to know the measure of angle “C” it would be formulated as follows: C = sin-1((c * sin(A)) / a).

However, this would apply only if you have the value of the side “c” to find the angle “C” or the value of angle “C” to find side “c”. If you do not have either of these two values, you would proceed to use the cosine theorem.

The law of cosines works for find the measure of a side or angle of a triangle Knowing the measures of the other two sides and the angle between them. An example of this would be, if we want to calculate the measure of side “a” of a triangle, we could use the cosine theorem as long as we know the sides “b” and “c” and the angle “A”.

In this way, the formula of the cosine theorem would be applied as follows: a^2= b^2 + c^2 – 2bc * cos(A).

However, if we do not have the value of the opposite angle on the side we are looking for, we can apply one of the formulas of the cosine theorem, isolating the value we are looking for.

### Problems that are solved with the law of sines

The Law of Sines is known to be very useful when you need to solve various problems involving triangles, especially when you need to determine the measures of its sides or angles. Next, we will describe some examples of the most common problems in which the law of sines is used:

1. Determine the length of an unknown side: The law of sines can be applied when you want to know the value of one side of a triangle. This as long as the value of the other two sides and the opposite angle of one of them are known.
2. Find the height of a triangle: if we know the length of a side of the triangle and its angle opposite to it, we can apply the law of sines. With it you can know the height of the triangle.
3. Solve Oblique Triangles: This type of triangle has angles that are not right. Knowing the measures of two of its sides and the opposite angle of one of them, we can use the law of sines to find the other two angles and the missing side.
4. Solve navigation problems: the law of sines is often used during navigation. This thanks to the fact that, with it, you can find the distance between two points on the earth’s surface.
5. Solve trigonometry problems in general: This theorem is one of the most important tools in trigonometry. It is often used to solve different problems involving triangles and angles.

These are the most common examples where it is applied the law of sines to solve problems. In general, it tends to be widely used to determine the measures of one side or angle of oblique triangles.

## Examples of the use of the law of sines in mathematics

Next, we will describe some examples or mathematical exercises where the law of sines applies:

Example 1: If we have a triangle whose sides are a=8 and b=7, knowing that angle A =45°, we can apply the law of sines to find angle B. To do this, we use the following formula: a/sin(A) = b/sin(B).

Solving, changing values ​​and solving we obtain that the value of angle B is 38.22°.

Example 2: Having a triangle with angles Alpha A=67° (opposite side a) and Beta B=36° (opposite side b), side a=6, the value of side c can be obtained. Since the sum of the internal angles of a triangle must always be 180°, we obtain that angle C=77°.

Now, to find the value of side ‘c’, apply the law of sines as follows: c/sin(C) = a/sin(A). Changing values ​​and isolating we have: c=(6* sin(77°))/sin(67°)with which we obtain that the value of the side c=6.35.