# Maths Class 11 Chapter 3. Trigonometric Functions

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1 1 P a g e Angle Maths Class 11 Chapter 3. Trigonometric Functions When a ray OA starting from its initial position OA rotates about its end point 0 and takes the final position OB, we say that angle AOB (written as AOB) has been formed. The amount of rotation from the initial side to the terminal side is Called the measure of the angle. Positive and Negative Angles An angle formed by a rotating ray is said to be positive or negative depending on whether it moves in an anti-clockwise or a clockwise direction, respectively. Measurement of Angles There are three system for measuring angles, 1. Sexagesimal System/Degree Measure (English System) In this system, a right angle is divided into 90 equal parts, called degrees. The symbol 1 is used to denote one degree. Each degree is divided into 60 equal parts, called minutes and one minute is divided into 60 equal parts, called seconds. Symbols 1 and 1 are used to denote one minute and one second, respectively.

3 3 P a g e Trigonometric (or Circular) Functions Let X OX and YOY be the coordinate axes. Taking 0 as the centre and a unit radius, draw a circle, cutting the coordinate axes at A,B, A and B, as shown in the figure. Now, the six circular functions may be defined as under (i) cos θ = x (ii) sin θ = y (iii) sec θ = 1/x, x 0 (iv) cosec θ = 1/y, y 0 (v) tan θ = y/x, x 0 (vi) cot θ = x/y, y 0 Domain and Range

4 4 P a g e Range of Modulus Functions sin θ 1, cos θ 1, sec θ 1, Cosec θ 1 for all values of 0, for which the functions are defined. Trigonometric Identities An equation involving trigonometric functions which is true for all those angles for which the functions are defined is called trigonometrical identity. Some identities are Sign of Trigonometric Ratios

5 5 P a g e Trigonometric Ratios of Some Standard Angles Trigonometric Ratios of Some Special Angles Trigonometric Ratios of Allied Angles Two angles are said to be allied when their sum or difference is either zero or a multiple of 90. The angles θ, 90 ± θ, 180 ± θ, θ, 360 θ etc., are angles allied to the angle θ, if θ is measured in degrees. Trigonometric Periodic Functions

6 6 P a g e A function f(x) is said to be periodic, if there exists a real number T> 0 such that f(x + T)= f(x) for all x. T is called the period of the function, all trigonometric functions are periodic. Maximum and Minimum Values of Trigonometric Expressions Trigonometric Ratios of Compound Angles The algebraic sum of two or more angles are generally called compound angles and the angles are known as the constituent angle. Some standard formulas of compound angles have been given below.

7 7 P a g e Transformation Formulae

8 8 P a g e Trigonometric Ratios of Multiple Angles

9 9 P a g e Trigonometric Ratios of Some Useful Angles

10 10 P a g e

11 11 P a g e

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