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It is used in many fields like game development, mechanical engineering, phonetics, medical imaging, architecture, seismology, civil engineering, electronics, oceanography, and chemistry number theory and even in image compression. There might be no direct applications of Trigonometry in everyday life, but a lot of its applications and concepts are used in various fields, for measurements, speed, directions, to estimate size, navigate directions etc.

Trigonometry is used to:

Ø  Measure the mountains: The trigonometric functions are used to measure the heights of mountains and hills. It is often used by civil engineers and architects to measure the heights of buildings.

Ø  In marine biology: It is often used by marine biologists to measure effects of sunlight on algae to photosynthesis. They also use Trigonometry to estimate the size of sea and ocean animals.

Ø  In aviation: Wind plays an important role on how a flight will travel. Which is why Trigonometric functions are used to measure speed, direction and distance of wind to lead the aeroplane in the right direction.

Ø  In navigation: The compass uses the Trigonometric concepts to show directions to travellers since ages. Which goes to show that Trigonometry has been one of the oldest concepts of Maths that was developed ages ago and used for years.

These are just some fields and there are a lot more that can be added to the list, like satellite systems to oceanography to even calculus where Trigonometric concepts are used regularly.

Listed below are the different topics of Trigonometry:

Differential calculus and differential equations

Analytic Trigonometry 

Euler's Formula

Ptolemy's Theorem

Pythagorean Theorem

 Differential Geometry 

Functions and Properties of Trigonometry

Trigonometric Equations 

Statistics and Probability 

 Trigonometry Project 

 Vector Calculus 

 Graphs of Trigonometric Functions 

Functions of Large and Negative Angles

 Trigonometric Identities - Reciprocal Identities

Half Angle formulas

Powers of complex numbers

Double Angle Identities

 Pythagorean Identities

Even and Odd Trig Functions

 Inverse Trigonometric Functions

Rational Trigonometry 

Trigonometric Functions 

Angle Definition and Properties of Angles

Introduction to the Six Trigonometric Functions

 Product to Sum and Sum to Product Formulas

 Quotient Identities

 Graphing the Cosine Function

Trigonometric Identities 

The Tangent Function in Right Triangles

 Logic 

 Calculating Trigonometric Functions 

Graph Theory 

Radian 

Interpolation

Reference Angles

Power Reduction Formula

Tensor Analysis 

 Law of Tangents 

 Sum and Difference of Angles Identities

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Yes, the three basic functions in trigonometry are sine, cosine and tangent. Based on these three functions the other three functions that are cotangent, secant and cosecant are derived. All the trigonometric concepts are based on these functions and if one understands this, and the basic Trigonometric formulas, it is easy to understand the concepts very efficiently. The various formulas in Trigonometry are:

Pythagorean Identities

sin²θ + cos²θ = 1

tan2θ + 1 = sec2θ

cot2θ + 1 = cosec2θ

sin 2θ = 2 sin θ cos θ

cos 2θ = cos²θ – sin²θ

tan 2θ = 2 tan θ / (1 – tan²θ)

cot 2θ = (cot²θ – 1) / 2 cot θ

Sum and Difference identities-

For angles u and v, we have the following relationships:

sin(u + v) = sin(u)cos(v) + cos(u)sin(v)

cos(u + v) = cos(u)cos(v) – sin(u)sin(v)

tan(u+v) = tan(u) + tan(v)1−tan(u) tan(v)tan(u) + tan(v)1−tan(u) tan(v)

sin(u – v) = sin(u)cos(v) – cos(u)sin(v)

cos(u – v) = cos(u)cos(v) + sin(u)sin(v)

tan(u-v) = tan(u) − tan(v)1+tan(u) tan(v)tan(u) − tan(v)1+tan(u) tan(v)

If A, B and C are angles and a, b and c are the sides of a triangle, then,

Sine Laws

a/sinA = b/sinB = c/sinC

Cosine Laws

c2 = a2 + b2 – 2ab cos C

a2 = b2 + c2 – 2bc cos A

b2 = a2 + c2 – 2ac cos B

THE BASIC TRIGONOMETRIC FUNCTIONS FORMULAS ARE:

·         sin θ = Opposite Side/Hypotenuse.

·         cos θ = Adjacent Side/Hypotenuse.

·         tan θ = Opposite Side/Adjacent Side.

·         sec θ = Hypotenuse/Adjacent Side.

·         cosec θ = Hypotenuse/Opposite Side.

·         cot θ = Adjacent Side/Opposite Side.

The 3 basic theorems of Trigonometry are:

        I.            The Pythagorean theorem:

It states that the square of the hypotenuse of a right triangle is the sum of the squares of its other two sides.

      II.            The Law of Sines:

This states the relationship between the sides and angles of non-right (oblique) triangles. .

    III.            The Law of Cosines

This states that the lengths of the sides of a triangle to the cosine of one of its angles. The 3 formulas of the Law of Cosines are:

·         a2 = b2 + c2 – 2bc cos α, where a,b, and c are the sides of the triangle and α is the angle between sides b and c.

·         b2 = a2 + c2 – 2ac cos β

·         c2 = b2 + a2 – 2ab cos γ