INVERSE TRIGONOMETRIC FUNCTIONS (UNIT - II)

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Syllabus:

Recapitulation of domain and range of sin x, cos x, tan x, cosec x, sec x and cot x and their graphs – Definition of inverse trigonometric functions-Domain and range of sin⁻¹x, cos⁻¹x, tan⁻¹x, cosec⁻¹x, sec⁻¹x, cot⁻¹x and their graphs – Principle values of inverse trigonometric functions – simple problems.

Function:

\[\text{A function f: A → B is a rule which assigns each element of A to a unique element in B.}\ \hspace{15cm}\\ \text{Here f(x) is called the image of the element x under the function ‘f’.}\ \hspace{15cm}\\ \text{A is called the domain and B is called the codomain of the function ‘f’.}\ \hspace{15cm}\]

Domain and Range of trigonometric functions:

\[\text{The domain of trigonometric function is set of angles or radians}\ \hspace{18cm}\\ \text{and the range is the set of real numbers}\ \hspace{15cm}\]

\[sin\ 30^0\ =\ \frac{1}{2}\ \hspace{10cm}\]

\[cos\ 0^0\ =\ 1\ \hspace{10cm}\]

\[tan\ 30^0\ =\ \frac{1}{\sqrt{3}}\ \hspace{10cm}\]

Inverse trigonometric functions:

\[\text{A function which is the reverse process of a trigonometric function is called the inverse trigonometric function}\ \hspace{5cm}\\ \text{sin⁻¹x, cos⁻¹x, tan⁻¹x, sec⁻¹x, cosec⁻¹x, cot⁻¹x are the inverse trigonometric}\ \hspace{15cm}\\ \text{functions of sin x, cos x, tan x, sec x, cosec x and cot x respectively.}\ \hspace{15cm}\]

\[If\ x\ =\ sin\ \theta,\ then\ \theta\ =\ sin^{-1}x\]

\[sin^{-1}\ (\frac{1}{2})\ =\ 30^0\ \hspace{10cm}\]

\[cos^{-1}\ (1)\ =\ 0\ \hspace{10cm}\]

\[tan^{-1}\ (\frac{1}{\sqrt{3}})\ =\ 30^0\ \hspace{10cm}\]

\[\text{The domain of inverse trigonometric function is set of real numbers and}\ \hspace{18cm}\\ \text{and the range is the set of angles or radians.}\ \hspace{15cm}\]

Graphs of function:

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\[\text{The set of all points (x, f(x)), x ∈ R determines the graph of the function f.}\ \hspace{15cm}\]

Amplitude of a graph:

\[\text{The amplitude is the maximum distance of the graph from the x-axis.}\ \hspace{15cm}\\ \text{Thus, the amplitude of a function is the height from the}\ \hspace{15cm}\\ \text{x-axis to its maximum or minimum.}\ \hspace{15cm}\]

\[\text{If y = A sin B x, then amplitude = |A|}\ \hspace{18cm}\]

Period of a graph:

\[\text{The period is the distance required for the function}\ \hspace{18cm}\\ \text{to complete one full cycle.}\ \hspace{15cm}\]

\[\text{If y = A sin B x, then period =}\ \frac{2\pi}{|B |}\ \hspace{18cm}\]

The Graph of sine function

x(in radian/	0	\[\frac{\pi}{6}\]	\[\frac{\pi}{4}\]	\[\frac{\pi}{3}\]	\[\frac{\pi}{2}\]	\[\pi\]	\[\frac{3pi}{2}\]	\[2\pi\]
degree)	\[0^0\]	\[30^0\]	\[45^0\]	\[60^0\]	\[90^0\]	\[180^0\]	\[270^0\]	\[360^0\]
y = sin x	0	\[\frac{1}{2}\]	\[\frac{1}{\sqrt{2}}\]	\[\frac{\sqrt{3}}{2}\]	1	0	– 1	0

\[\text{y = sin x is periodic with period}\ 2\pi.\ \hspace{18cm}\]

\[\text{The portion of the curve corresponding to 0 and}\ 2\pi\ is\ called\ \hspace{15cm}\\ \text{a ‘cycle’. Its amplitude is 1.}\ \hspace{18cm}\]

The Inverse sine function:

\[\text{The inverse sine function sin⁻¹: [-1 , 1] →}\ [ -\ \frac{\pi }{2}\ ,\ \frac{\pi }{2}]\ \text{such that sin y = x}\ \hspace{15cm}\]

Principal Value:

\[\text{The restricted domain}\ [ -\ \frac{\pi }{2}\ ,\ \frac{\pi }{2}]\ \text{is called the principal domain of the sin function}\ \hspace{10cm}\\ \text{and the values of y = sin⁻¹x, -1 ≤ x ≤1, are known as principal values of the function y = sin⁻¹x}\ \hspace{10cm}\]