9.1 Q-8

Exercise 9.1 — Question 8

Problem

Find the domain and range of the following trigonometric functions, and state whether each is even, odd, or neither. Also state the period of each function:

(i) $y=\sin x$    (ii) $y=\cos x$    (iii) $y=\tan x$

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Solution

(i) $y=\sin x$

(ii) $y=\cos x$

(iii) $y=\tan x$

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Key Concepts

Even Function: A function $f$ is even if $f(-x)=f(x)$ for all $x$ in its domain. Its graph is symmetric about the $y$-axis.

Odd Function: A function $f$ is odd if $f(-x)=-f(x)$ for all $x$ in its domain. Its graph has rotational symmetry about the origin.

Periodic Function: A function $f$ is periodic with period $T$ if $f(x+T)=f(x)$ for all $x$ in its domain, where $T>0$ is the smallest such value.

• $\sin (x+2\pi )=\sin x$  →  period $=2\pi$
• $\cos (x+2\pi )=\cos x$  →  period $=2\pi$
• $\tan (x+\pi )=\tan x$  →  period $=\pi$