Solution of Triangles

A triangle has six characteristics of which three are the angular characteristics (angles) and the rest three are the linear characteristics (sides). These characteristics under fixed conditions make up a triangle. Even if only half of these conditions are known to us, it is possible to make a complete triangle by using Sine, Cosine or trigonometric functions.

In a triangle, these factors are dependent on each other to a certain degree. Hence, their computation is possible even if only a few others are known to us. 

Main trigonometric problem

The greatest problem that arises during the computation of all the characteristics of a triangle is the actual measure of their characteristics utilising trigonometric properties alone. It will significantly mean that it is not likely to find out the characteristics of a triangle solely on the basis that three angles of a triangle are given. 

It is a mandatory requirement to have at least a single side known to accurately calculate the solution is a triangle. If only three angles for a triangle are given to us, then the conditions will stand true for every triangle that has the same angles. It will not give an accurate measure of the linear characteristics (sides) of the given triangle. 

However, on the other hand, it is possible to give a brief account of the characteristics of a triangle based on three sides alone. This measurement works on the basis that the measure of an angle of a triangle is a result of the relation between two sides of the triangle. Hence, it is possible to find the solution of a triangle using linear characteristics alone, while the same does not stand true for angular characteristics.

Finding the characteristics

To find the solutions of a triangle, we must be aware of at least three characteristics of the triangle to be known, and one of those characteristics must be a linear characteristic. 

Following are the methods that can be used to compute the characteristics of a triangle:

Assume there to be a triangle ABC. Let the three linear characteristics (sides) of the triangle be “a, b, c”, and let the angular characteristics (angles) of the triangle be “α, β, γ”.

1. When three sides of a triangle are given (SSS)

When the three sides of a triangle are known, it is possible to calculate the angles of the triangle using the Cosine formula. 

This formula is as follows:

CosA = (b² + c² – a²) / 2bc

CosB = (a² + c² – b²) / 2ac

CosC = (a² + b² – c²) / 2ab

2. When two sides and an angle in between are given (SAS)

When two sides and an angle of a triangle are known, it is possible to compute the third side using the cosine rule. After that angles can be found in the same way using the sum of angle properties. 

The Cosine rule is as follows:

a² = b² + c² – 2bcCosα

b² = a² + c² – 2acCosβ

c² = a² + b² – 2abCosγ

Sum of angle property:

α + β + γ = 180°

3. When two sides and a non-included angle are given (SSA)

In this case, the sine formula is used to figure out the rest of the characteristics related to the triangle:

The sine formula is as given below:

a / Sinα = b / Sinβ = c / Sinγ

4. When a side and two adjacent angles to that side are given (ASA)

In this case, it is possible to find out the remaining two sides and an angle using the sine rule. 

5. When three angles of a triangle are given (AAA)

We can find out the measure of sides of a triangle using trigonometric formulas over the given sides. However, it is to be noted that this result will stand true for all the similar triangles. This means, the triangles that have their sides in proportion to this triangle, will have the same angles.

Angles and lengths of sides

To determine the characteristics and solution of a triangle it is to be noted that there are a certain set of rules that are to be followed in this regard. A few of those rules are:

1. Sum of angles

A triangle as its name states has three angles. The sum of all three interior angles of a triangle makes a total of 180°. If the sum of internal angles of a three-sided closed figure is not 180°, then it won’t be considered a triangle.

Also, in a right-angled triangle, one angle is 90°, so the sum of the other two angles needs to be 90° too.

2. The measure of sides of a triangle

It is to be noted that the sum of any two sides of a triangle is always greater compared to the third side of the given triangle. If a figure does not stand true to that property, then it is not a triangle.

In the case of a right-angled triangle, when we take the square of the sum of the base and height of a triangle, it will always equate to the square of the hypotenuse of that right-angled triangle. 

H² = B² + P²

Here, 

H = Hypotenuse

B = Base

P = Perpendicular

3. Area of a triangle

If the three sides of a triangle are known, it is possible to calculate the area of that triangle. 

Area = [s(s-a)(s-b)(s-c)]^1/2

where “s” stands for semi-perimeter,

And, a,b and c are the sides of the triangle.

4. Trigonometric properties

For right-angled triangles, computation of solution is much easier. As one of the characteristics, the 90° angle, is already known. It is possible by using trigonometric functions over that triangle. 

Conclusion

It is possible to find out the solution of a triangle if only three of its characteristics are known. By utilising the characteristics of the triangle that are known to us, it is possible to find out other characteristics. And, with that the study of all the properties of the given triangle is possible. 

When all the characteristics of a triangle are known, we can draw out all the trigonometric and functional properties of a triangle. We can use these properties to find out the projection or area of the triangle.