Difference between Trigonometry and Geometry

Mathematics can be divided into three big branches. They are algebra, geometry and arithmetic. Geometry talks about shapes, size, position and space. Trigonometry is a subset of geometry. Trigonometry just involves triangle shapes. Although it can be used in other shapes, but in using trigonometry, we separate the shapes into some triangles.

In the early time, geometry developed by the Greek’s mathematician Euclid. This elementary geometry involves two and three-dimensional (plane and solid geometry), known as Euclidean geometry. The coordinate system proposed by Rene Descartes and the development of algebra introduce new concept in geometry called analytic geometry. This analytic geometry represents a curve in an algebraic equation or functions.

The first use of trigonometry was about 150 BC. Hipparchus, the Hellenistic mathematician, was the one who use it. He compiled a trigonometric table using the sine in solving triangles. In the ancient world, trigonometry was used in sailing as a navigation method. Many works from talented men helped trigonometry develop to a sophisticate subject as it is now.

The area of a plane is a one studied in geometry. There are many formulas can be used to find the area of planes in any shapes. The area of a rectangular is the product of its length and width. The area of a parallelogram is the product of its height and base, and so on.

In trigonometry, we study about the proportion of the side length of a triangle. There are three basic proportionalities in trigonometry. They are sine, cosine and tangent. Suppose we have a right triangle. We pay attention to one of its acute angles. Suppose we call this angle theta. The longest side in a right triangle is called hypoteneuse. The side in front of the angle theta is called opposite side. The side left behind is called adjacent side. Then the sine of the angle theta is defined as the length of the opposite side divided by the length of the hypotenuse. The cosine of the angle theta is defined as the length of the adjacent side divided by the length of the hypotenuse. The tangent of the angle theta is defined as the length of the opposite side divided by the length of the adjacent side.

There are three more proportionalities in addition to the three basic ones discussed above. They are secant, cosecant and cotangent. They are the reciprocal of cosine, sine and tangent respectively.

Suppose there is a right triangle with perpendicular sides given. We can calculate its area easily. The area is one half the products of its perpendicular sides. Finding the area of a triangle is a geometry problem. Suppose the triangle is an acute triangle in spite of the perpendicular one. This is a triangle whose the three angles are less than 900. Assume we know the length of the three sides. How to work its area out? We have to use trigonometry. I mean, to find the height of the triangle we can use sine.

Assume we have an acute triangle ABC. We denote the side in opposite with the angle A as a. The side in opposite with angle B is denoted as b. Moreover, we denote the side in opposite with the angle C is c. If we take side AB as the base of the triangle then the height of the triangle will be the product of b and sine A. The height can also be the product of a and sin B. Then the area of the triangle is calculated using the geometry area formula. That is the product of the base (c) and the high (b x sin A) or (a x sin B) divided by 2.

There are many times when we have geometry problems, we can solve it without using trigonometry. Finding the volume of a cube, cylinder, and any other three dimensional object can be done with geometry formulas. We can use simple formula to find distance between a point and a line in a cube. Suppose we have a cube name ABCD EFGH. To find the distance from point H to the line AC we can use Pythagorean principle. To find the angle between the line AH and the plane ABCD we need trigonometry. So the trigonometry is used in a geometry problem.

By considering the example above, we can conclude that trigonometry differs from geometry in some reasons. Geometry is a superset of trigonometry. Geometry is a main branch in mathematics. Trigonometry is a useful and helpful tool to solve many geometry problems.