Difference Between Trigonometry and Geometry for JEE Main 2024

Mathematics has various important branches such as algebra, geometry, probability, trigonometry, arithmetic, and lots more. We know that geometry is the study of different shapes, sizes, and positions of different shapes based on the number of sides, angles, and so on. Whereas trigonometry is the subset of geometry that deals with the properties of one of the shapes in geometry called “triangle”. Both trigonometry and geometry seem to be related to each other, but they are not the same. 

In this article, let us discuss the difference between geometry and trigonometry with some detailed explanation.

What are Trigonometry and Geometry?

The branch of mathematics in which the relationship between the angles and the ratios of the sides of the right-angled triangles is studied is known as trigonometry. Ratios used to study these relationships which include, tangent, sine, cosine, secant, cotangent, and cosecant are known as the trigonometric ratios. We use trigonometry in order to find the unknown dimensions of any right-angled triangle by using the formulas and the identities. 

The branch of mathematics in which the principles related to the angles, patterns, covered distances, areas, and volumes, is studied is known as geometry. Geometry includes the study of all the concepts related to spatial and visual. Geometry can be classified into three types- euclidean, elliptical, and hyperbolic. 

The geometry in which we study the properties of a planar surface and solid figures which are based upon theorems and axioms is known as Euclidean geometry. While the geometry in which does not hold Euclid’s parallel postulates is known as elliptical geometry. The geometry in which we study hyperbolic surfaces is known as hyperbolic geometry.

Let us have a look at the difference between geometry and trigonometry.

What is  the Difference Between Trigonometry and Geometry?

• Trigonometry

Trigonometry can be considered as a subset of geometry. In modern mathematics, trigonometry plays a huge role. Trigonometry is mostly about studying various properties of triangles, lengths, and angles. However, it also deals with waves and oscillations. 

In trigonometry, we mostly study the relationships between the side lengths and angles of a right-angle triangle. There are six trigonometric relations. Three basic ones, named Sine, Cosine, and Tangent, are clubbed together with Secant, Cosecant, and Cotangent. 

Let us consider that we have a right-angled triangle. The three sides will be height, base, and hypotenuse respectively. Then we can define the fundamental trigonometric relations as follows:

sin A=(height)/(hypotenuse)

cos A=(base)/(hypotenuse)

tan A=(height)/(base)

Cosec, Sec, and Cot can also be stated as the reciprocal of Sine, Cosine, and Tangent respectively. Trigonometry is not just a study of simple plane figures. It has a branch called spherical trigonometry, which studies triangles in three-dimensional spaces.

• Geometry 

Geometry is defined as the study of various sizes, shapes, and properties of vacant spaces of a given number of dimensions such as 2D or 3D. Euclid, the great mathematician, had made a huge contribution to the domain of geometry. Therefore, he is known as the Father of Geometry. Geometry can be categorized into the following -

• Plane geometry, 

• Solid geometry, and 

• Spherical geometry. 

Plane geometry deals with two-dimensional geometric objects such as points, lines, curves, and various plane figures such as circles, triangles, and polygons. Solid geometry studies three-dimensional objects such as various polyhedra, spheres, cubes, prisms, pyramids, and so on. Spherical geometry also studies three-dimensional objects such as spherical triangles and spherical polygons. 

Geometry can also be categorized into Euclidean Geometry, the study of flat surfaces, and Riemannian geometry in which the fundamental subject is the study of curved surfaces.

Trigonometry and Geometry: Difference

Solved Examples

• Trigonometry

1. Prove \[\frac{\left (\text{1 - sin A}  \right )}{\left (\text{1 + sin A}  \right )} = \left ( \text{sec A - tan A} \right )^{2}\]

Solution- Taking left-hand side and multiplying both the numerator and denominator by (1 - sin A)

\[\frac{\left (\text{1 - sin A}  \right )}{\left (\text{1 + sin A}  \right )} = \frac{\left (\text{1 - sin A  }\right )^{2}}{\left ( \text{1 - sin A} \right )-\left ( \text{1 + sin A} \right )}\]

Since sin² θ + cos²θ = 1 ⇒ cos²θ -1 = sin²θ

\[\frac{\left (\text{1 - sin A}  \right )}{\left (\text{1 + sin A}  \right )} = \frac{\left ( \text{1 - sin A} \right )^{2}}{\left ( 1 - sin^{2}A \right )}\]

\[\frac{\left (\text{1 - sin A}  \right )}{\left (\text{1 + sin A}  \right )} = \left \{ \frac{\text{1 - sin A}}{\text{cos A}} \right \}^{2}\]

\[\frac{\left (\text{1 - sin A}  \right )}{\left (\text{1 + sin A}  \right )} = \left ( \frac{1}{\text{cos A}}-\frac{\text{sin A}}{cos A} \right )^{2}\]

\[\frac{\left (\text{1 - sin A}  \right )}{\left (\text{1 + sin A}  \right )} = \text{sec A - tan A}^{2}\]

Hence proved.

2. Consider that two ships are sailing on the opposite sides of a lighthouse. The angle of elevation of the highest point of the lighthouse 100m high, when observed from the ships, is 30º and 45º respectively. Find the distance between the two ships.

Ans: Construct a triangle ACD where B is a point on AC and BD be the height of the lighthouse and A and C are the positions of the ships. Then, 

BD = 100 m, angle BAD = 30°, angle BCD = 45°

tan 30° = BD/BA 

or, BA = 100 √3m

tan 45°= BD/BC 

or, BC = 100m

Distance between two ships = BA + BC = 100 (1+√3) = 273m.

• Geometry

1. Consider ABCD is a parallelogram such that AB is parallel to DC and DA is parallel to CB. The side AB is of length 20 cm. E is a point between A and B such that AE is 3 cm. F is a point in between  D and C. Find the length of DF such that EF divides the parallelogram into two parts with exactly equal areas.

Ans: Draw the trapezoid.

Let the area of the trapezoid AEFD be A1. 

A1 = (1/2) h (AE + DF) 

Let ‘h’ be the height of the parallelogram.

Suppose the area of the trapezoid EBCF be A2. 

A2 = (1/2) h (EB + FC) 

We know,

EB = 20 - 3 = 17 , FC = 20 - DF

We should now substitute EB and FC in A2 = (1/2) h (EB + FC)

A2 = (1/2) X (h) X (17 + 20 - DF) 

We need to have two equal areas A1 and A2 for EF to divide the parallelogram, 

(1/2) X (h) X (3 + DF) = (1/2) X h X (37 - DF)

On multiplying both sides by 2 and then dividing them by h,

3 + DF = 37 - DF

Solve for DF

2DF = 37 - 3

2DF = 34

DF = 17 cm

Did You Know?

1. The term “Geometry” is derived from Greek, where “Geo” refers to “Earth” and “metron” means “measure”.

2. The Greek words "trigonon" and "metron" when clubbed together form the term “Trigonometry”. Hipparchus, a Greek mathematician invented trigonometry.