# Solution of Triangles Using Trigonmetry

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Trigonometry is the study of triangles. In any triangle the $3$ sides and the $3$ angles are often called the elements of the triangle.

Trigonometry is the study of triangles. In any triangle the $3$ sides and the $3$ angles are often called the elements of the triangle.

In any triangle $\triangle ABC$. the side $BC$, opposite to the angle $\angle A$, is denoted by $a$; the sides $CA$ and $AB$ opposite to the angles $\angle B$ and $\angle C$ respectively, are denoted by $b$ and $c$.

When some of the elements of a triangle are given, the process of calculating its other unknown elements is called the Solution of the Triangle.

In a triangle, when the $3$ angles are given, only the ratios of the lengths of the sides can be found, so that the triangle is given in shape only.

For better illustration, suppose that three angles of a triangle are $51.34^\circ$, $68.2^\circ$, and $60.46^\circ$.

In the above triangles $\triangle ABC$ and $\triangle DEF$, we have

$\fracABDE$ $=$ $\fracBCEF$ $=$ $\fracCAFD$ $=$ $\frac12$.

This shows that we can construct infinitely many equiangular triangles and they all are of the same shape and the ratio of the corresponding sides is same.

We have learned from elementary geometry that a triangle is determined when we know any three of its parts (sides and angles), at least one of them being a side. These data enable us to construct the triangle; but elementary geometry does not teach us how to calculate the remaining parts. The reason is that sides and angles are expressed in different units.

The primary object of the science of Plane Trigonometry is to develope a method of solving plane similar triangles. Note that it is possible to establish relations between the sides $a$, $b$, and $c$ and the trigonometrical ratios of the angles $\angle A$, $\angle B$, and $\angle C$ of a triangle $\triangle ABC$.

These relations are very useful in solving a triangle, when

1. All the three sides are known,
2. Two sides and the included angle are known,
3. Two sides and the angle opposite to one of them are known, or
4. One side and two angles are known.

In case of a right-angled triangle,one of the angles is a right-angle ($90^\circ$). As the sum of three angles in a triangle is $180^\circ$, if we know one of the acute angles, we can obtain the other acute angle by subtracting it from $90^\circ$. Further, if we know two sides of a right-angled triangle, we can find the other side by using the Pythagorean Theorem. So, we can solve a right-angled triangle, when we know

1. Any two sides.
2. One of the sides and an acute angle.

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