Trigonometry-Properties of Triangle-Solution of Triangles

Introduction

Trigonometry is considered one of the most necessary and important branches of mathematics. It holds the solution of triangles (ambiguous cases). It involves understanding the concept of sides and angles of a right-angled triangle and their relationship. This theory was first given by the Greek mathematician Hipparchus.

Trigonometry

Trigonometry is a branch of mathematics with vast applications in several fields. “Trigonometry” acts with the concept of sides and angles of a right-angled triangle and the relationship between them. Missing, unknown angles and sides of any right-angled triangle can be derived with the help of some trigonometric formulas and functions. The measurements of angles are done in either degree or radians in trigonometry. The commonly used angles of trigonometry are 0°, 30°,45°,60° and 90°.

It can be further divided into two parts:-

• Plane trigonometry.
• Spherical trigonometry.

Trigonometric Table:

Uses of Trigonometry-solution of triangles (ambiguous cases)

Over the entire course of time, geometry has been applied in regions like engineering, divine mechanics, reviewing, and so forth. Its applications include:

• Different fields like oceanography, seismology, meteorology, actual sciences, stargazing, acoustics, route, hardware, etc.
• Tracking the distance of long streams, measuring the tallness of mountains, and so on.
• Round geometry has been utilised to find sunlight-based, lunar, and heavenly positions.

Properties of Triangles

Properties of a triangle depend on its sides and angles. We know that a triangle is a closed polygon comprising three sides and three vertices. The sum of each of the three inward angles of a triangle is equivalent to 180°.

Properties of a Triangle 

 Angle Sum Property

According to the angle sum property, the amount of the three inside angles of a triangle is 180° all of the time.

In the given triangle, ∠P + ∠Q + ∠R = 180°

Pythagoras Property

According to the Pythagoras hypothesis, in a right-calculated triangle, the square of the hypotenuse is equivalent to the sum of the squares of the other different sides. Numerically, it can be written as Hypotenuse² = Base² + Altitude². 

Congruence Property

According to the Congruence Property, two triangles are supposed to be harmonious if their comparing sides and points are equivalent.

∠XYZ = ∠DEF

∠YXZ = ∠EDF

∠YZX = ∠EFD

XY = DE

XZ = DF

YZ = EF

Some properties of the solution of triangles (ambiguous cases):

1. The total sum of the multitude of triangle angles (of different types) is equivalent to 180°.
2. The length of the different sides of a triangle is more prominent than the length of the third side.
3. The contrast between the different sides of a triangle is not exactly the length of the third side.
4. The side inverse, the more prominent angle, is the longest side of the multitude of three sides of a triangle.
5. The outside point is equivalent 100% of the time to the inside inverse points. This property of a triangle is called an exterior point property.
6. Two triangles are supposed to be comparative if the relating angles of the two triangles are congruent, and the lengths of their sides are proportional to each other.
7. Area of a given triangle = ½ × Base × Height
8. The perimeter of a triangle = amount of all its three sides.

Solution of Triangles (ambiguous Cases)

The “Ambiguous Case” (SSA) happens when we are given different sides and the inverse of the angle one of these given sides. The solution of triangles (ambiguous cases) occurring because of this condition should be investigated considerably more intently than the SSS, ASA, and AAS cases, because SSA might bring about one triangle, two triangles, or even no triangle.

Solution of Triangles (ambiguous cases) using Law of Sines

This includes the following stages:

1. Using The Law of Sines first to compute one of the other two angles;
2. Using the three points, add to 180° to find out the other angles;
3. Utilise The Law of Sines again to see the obscure side.

Conclusion

The sine rule of solution of triangles (ambiguous cases) is used when given either a) two angles and one side, or b) different sides and a non-included angle. The cosine rule is utilised when given either a) three sides or b) different sides and the included angle. The use of SSS, ASA, or AAS always ensures the creation of a single, unique triangle. Solution of triangles (ambiguous cases) happens when we are given two sides and the angle opposite one of these two sides.