EXAMPLE 11

Sketch the graph of y = 10 sin x + 2 sin (3x + p/2). Find the period, maxima and minima.

Solution:

The function y = 10 sin x + 2 sin (3x + p/2) has

(i) Period = 2p
(ii) Maximum = 10.5 at maxima = 102⁰
(iii) Minimum = -10.5 at minima = 282⁰

EXAMPLE 12

Find the period, phase shift and the location of the asymptotes for the function  

Solution :

i) The period of the function is obtained by

ii) The phase shift of the function is given by since the phase shift is positive, it is to the left.

iii) Now the asymptotes can be found by solving (bx + c) =   and (bx + c) = - as

Note I

i) The general equations of tangent and cotangent functions are-

y = d + a tan (bx + c) and y = d + a cot (bx + c)

ii) \ Period = p / b

iii) The phase shift = |c / b| when c/b < 0 then the shift is to the right and when c / b > 0 then the shift is to the left.

iv) Asymptotes are obtained on solving bx + c = p /2 and bx + c = -p /2.

v) Since tangent and cotangent are unbounded the variable 'a' does not represent "amplitude".

vi) The vertical shift is represented by "d" .

Note II     For Secant and Cosecant functions

i) Amplitudes are not defined.

ii) As sec and csc are reciprocals of sin and cos, they have the same period (2p/b) and phase shift = | c / b |.

iii) Asymptotes of these functions are obtained by solving the two equations bx + c = 0 and bx + c = 2p.

Index

5. 1 Circular function
5. 2 Periodic function
5. 3 Even & Odd
5.4 Graphs of Trigonometric Functions Supplementary Problems

Chapter 6