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Mathematics and Statistics: Trigonometric Functions

Topics are arranged as per SYLLABI published by Maharashtra State Board of Secondary and Higher Secondary Education.

Trigonometry is measurement triangles.
Measurement of sides, angles and various relations exists between them.
In broad categories there are two types of Trigonometric Functions:

  • Circular : They are used to study periodic phenomena (Ex: sin x )
  • Hyperbolic : They are analogous to Circular Functions (Ex: sinh x )
For the scope of our study, we’ll focus only on Circular Functions.
Three mains trigonometric functions can be studied with the help of "Right angled Triangle" There are three are three main trigonometric functions.
  • Sine (sin) = Opposite / Hypotenuse
  • Cosine (cos) = Adjacent / Hypotenuse
  • Tangent (tan) = Opposite / Adjacent = Sine / Cosine

Three important functions are best defined using the above three main functions.
  • Secant (sec) = 1 / Cosine = Hypotenuse / Adjacent
  • Cosecant (cosec) = 1/ Sine = Hypotenuse / Opposite
  • Cotangent (cot) = 1 / Tangent = Adjacent / Opposite

Fundamental identities: [Note: Sine, Cosine and Tangent are all based on a Right-Angled Triangle.]
Trignometric Functions where Trignometric Functions
Trignometric Functions where Trignometric Functions
Trignometric Functions where Trignometric Functions

  • The equation that involves a trigonometric ratio of variables is called a Trigonometric Equation.
  • The values of variables present in trigonometric equation for which the equation is satisfied is called the solution of that trigonometric equation.
  • There can be infinitely many values that satisfy trigonometric equation, but out of these values those which lie in the interval Trignometric Functions are called the principal solutions.
  • If solutions of a trigonometric equation are generalized by using its periodicity and represented by an equation or a formulae, they are called the general solutions of the trigonometric equations.

Trigonometric Equation General Solution Key Points
Trignometric Functions Trignometric Functions
Trignometric Functions Trignometric Functions
Trignometric Functions Trignometric Functions General solution is same as that of Trignometric Functions
Trignometric Functions = Trignometric Functions Trignometric Functions; Trignometric Functions
Trignometric Functions = Trignometric Functions Trignometric Functions; Trignometric Functions
Trignometric Functions = Trignometric Functions Trignometric Functions; Trignometric Functions
Trignometric Functions = Trignometric Functions Trignometric Functions; Trignometric Functions
Trignometric Functions = Trignometric Functions Trignometric Functions; Trignometric Functions
Trignometric Functions = Trignometric Functions Trignometric Functions; Trignometric Functions
Trignometric Functions Trignometric Functions Trignometric Functions

Polar coordinates: Polar and Cartesian co-ordinates of a point are related. Trignometric Functions
Where Trignometric Functions and Trignometric Functions for Trignometric Functions
In polar coordinate system; we’ve fixed point called as pole. It’s equivalent to origin in Cartesian co-ordinate system.

  • Triangle can be expressed in three parts.
  • The three sides and three angles of a triangle are together known as parts of the triangle.
  • If at least three of them are known (atleast one is its side), then the rest three can be obtained. The process of obtaining unknown parts of a triangle is called as solving the triangle or solution of triangle.
Let’s assume there is a triangle ABC having side BC = a, CA=b and AB=c . Let’s go through various rules and theorems for solving triangles.

Sides of a triangle are proportional to the sines of the opposite angles.
In the triangle ABC: Trignometric Functions

In the triangle ABC

  • Trignometric Functions
  • Trignometric Functions
  • Trignometric Functions

In the triangle ABC

  • a = b cosC + c cosB
  • b = c cosA + a cosC
  • c = a cosB + b cosA

Let’s apply Sine Rule, Cosine Rule and Projection Rule to derive Half angle formula. In triangle ABC [Note: 2s = a + b + c ]

Trignometric Functions Trignometric Functions Trignometric Functions
Trignometric Functions Trignometric Functions Trignometric Functions
Trignometric Functions Trignometric Functions Trignometric Functions

If a, b, c are the lengths of the sides BC, CA and AB of a triangle ABC such that 2s = a + b + c then the area of the triangle ABC = Trignometric Functions

In triangle ABC

  • Trignometric Functions
  • Trignometric Functions
  • Trignometric Functions

Function Trignometric Functions Domain of existence Range of Trignometric Functions Inverse of Function Trignometric Functions Principal Value of the range Quadrant
y = sinx Trignometric Functions [-1,1] Trignometric Functions Trignometric Functions, Trignometric Functions Fourth, First
y = cosx Trignometric Functions [-1,1] Trignometric Functions Trignometric Functions, Trignometric Functions First, Second