Question 3, Exercise 9.1
Solutions of Question 3 of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.
Question 3(i)
Find domain and range: $y=7 \cos 4x$
Solution.
AS \begin{align*} & -1\leq \cos 4x \leq 1 \,\, \forall \,\, x\in \mathbb{R} \\ \implies & -7\leq 7 \cos 4x \leq 7 \\ \end{align*} Thus domain $= ]-\infty, \infty[ = \mathbb{R}$
Range $=[-7,7]$.
Question 3(ii)
Find domain and range: $y=\cos \frac{x}{3}$
Solution.
AS \begin{align*} & -1\leq \cos \frac{x}{3} \leq 1 \,\, \forall \,\, x\in \mathbb{R} \\ \end{align*} Thus domain $= ]-\infty, \infty[ = \mathbb{R}$
Range $=[-1,1]$.
Question 3(iii)
Find domain and range: $y=\sin \frac{2 x}{3}$
Solution.
AS \begin{align*} & -1 \leq \sin \frac{2x}{3} \leq 1 \,\, \forall \,\, x \in \mathbb{R} \end{align*} Thus domain $= ]-\infty, \infty[ = \mathbb{R}$
Range $=[-1, 1]$.
Question 3(iv)
Find domain and range: $y=7 \cot \frac{\pi}{2} x$
Solution.
Let $\theta=\frac{\pi}{2} x$. Then $$y=7 \cot \theta$$
Domain of $y=\left\{\theta: \theta\in \mathbb{R} \text{ and } \theta \neq n\pi, n\text{ is integer} \right\}$
Range of $y=\mathbb{R}$
As \begin{align*} & \theta \neq n\pi \\ \implies & \dfrac{\pi}{2} x \neq n\pi \\ \implies & x \neq 2n \end{align*}
Hence domain of $y=\left\{x: x\in \mathbb{R} \text{ and } x \neq 2n, n\text{ is integer} \right\}$
Range of $y=\mathbb{R}$.
Question 3(v)
Find domain and range: $y=4 \tan \pi x$.
Solution.
Let $\theta=\pi x$. Then $$y=4 \tan \theta$$
Domain of $y=\left\{\theta: \theta\in \mathbb{R} \text{ and } \theta \neq (2n+1)\frac{\pi}{2}, n\text{ is integer} \right\}$
Range of $y=\mathbb{R}$
As \begin{align*} & \theta \neq (2n+1)\frac{\pi}{2} \\ \implies & \pi x \neq (2n+1)\frac{\pi}{2} \\ \implies & x \neq \frac{2n+1}{2} \end{align*}
Hence domain of $y=\left\{x: x\in \mathbb{R} \text{ and } x \neq \frac{2n+1}{2}, n\text{ is integer}\right\}$
Range of $y=\mathbb{R}$.
Question 3(vi)
Find domain and range: $y=\operatorname{Cosec} 4 x$
Solution.
Let $\theta=4 x$. Then $$y= \operatorname{Cosec} \theta$$
Domain of $y=\left\{\theta: \theta\in \mathbb{R} \text{ and } \theta \neq n\pi, n\text{ is integer} \right\}$
Range: $y\leq -1 \text{ and } y\geq 1$.
As \begin{align*} & \theta \neq n\pi \\ \implies & 4 x \neq n\pi \\ \implies & x \neq \frac{n \pi}{4} \end{align*}
Hence domain of $y=\left\{x: x\in \mathbb{R} \text{ and } x \neq \frac{n \pi}{4}, n\text{ is integer} \right\}$
Range: $y\leq -1 \text{ and } y\geq 1$.
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