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This is one of our many quizzes on Interpreting trigonometric graphs. Click on this to access more such competitive quizzes on Trigonometry.

Interpreting trigonometric graphs Quiz 2

mathematics

Sachin went to a fair in his town. He took a ride on a vertical merry-go round which is circular in shape. His height from the ground with time is shown in the below graph. Tangential speed of Sachin is <img src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/a412fd50-2c46-46b8-8ee4-a20d8569e68d6979869103378281958.png"/>

 To find tangential speed, distance and time for one rotation of the merry go round is needed. Try to find the time for one revolution and to find distance and find the radius of the merry go round from the graph given.

From the graph it is clear that the difference between maximum height and minimum height $=25-5=20\text{ m}$ . This difference will be equal to the diameter of the merry-go round. So, radius of the merry go round $=10\text{ m }\text{.}$ Also time taken for one rotation $=10\text{ sec}$ . So, tangential speed $=\dfrac{2\pi \times \text{radius}}{\text{time for one rotation}}=\dfrac{2\pi \times 10}{10}=2\pi \text{ m/s}$

 $\text{The graph for }\left| \max \left\{ \sin x,\cos x \right\} \right|$ is 

 Draw the graph for $y=\cos x$   and $y=\sin x$. Then find where one out of two is maximum. From the graph obtained, the curve on the negative side of the ordinate is rotated about abscissa keeping the graph on the positive side of the ordinate as it is. Then analyze the graph according to the question.

<img src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/2cf35b67-7688-451d-aa09-6b7769f0d5172086298805527212785.png"/> <img src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/7b2ea0f1-c9a0-4f2a-b978-025a6ba73bb74815676682015000747.png"/> To make the graph for $\max \left\{ \sin x,\cos x \right\}$ , draw the curves for $y=\cos x$ and $y=\sin x$. Draw the curves where one out of $y=\cos x$ and $y=\sin x$ is maximum. Then we obtain the above graph. To draw the graph for $\left| \max \left\{ \sin x,\cos x \right\} \right|$ , the curve on the negative side of the ordinate is rotated about the abscissa keeping graph on the positive side of the ordinate as it is as given below. <img src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/37bbdb2d-08a2-407c-9f7b-8c9e77217a9c155045544292920583.png"/> So, the time period of the above graph $=2\times \left( \frac{5\pi }{4}-\frac{\pi }{4} \right)=2\pi $

 The number of solution(s) of the equation $\cos 2x=\tan x$  in the interval $\left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]$  is/are

 Draw the curve for $y=\cos 2x\text{ and }y=\tan x$ . Then find the common points between these two curves to find the given number of solutions.

<img src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/31c72e9a-79e6-48ef-81e4-1fda06ae8f393129546355848680375.png"/> We have drawn the curve for $y=\cos 2x\text{ and }y=\tan x$ . There is one common point between these two curves. Therefore, there is one solution of the equation $\cos 2x=\tan x$ in the interval $\left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]$.

Find the value of $a+b$ for the function of the form $f\left( x \right)=cosec\left( ax+\dfrac{\pi }{2} \right)+b$ whose graph is given below: <img src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/25ee7ef1-da11-414a-9167-8486672fea0e7633648854226871755.png"/>

 Identify the time period from the figure given in the question to find the value of $a.$ Find the shift in the curve in $y-axis$  to find the value of $b$.

From the graph time period is $\pi .$ $\Rightarrow a=2$ Also the graph is shifted in negative $y-axis$ by 2 units. $\Rightarrow b=-2$ $\therefore a+b=2+\left( -2 \right)=0$

 The range of the function $f\left( x \right)=\sin \left\{ x \right\}\text{ is, where }\left\{ .\text{ } \right\}$ represents the fractional part function.

 The range of fractional part functions is the domain for $f\left( x \right)=\sin \left\{ x \right\}$. Hence use the range of fractional part function to find the range for $f\left( x \right)=\sin \left\{ x \right\}$.

<img src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/1b966bd8-edbc-4aa2-8ebd-f570ba07a0a42086202824147339492.png"/> For any interval $\left[ n,n+1 \right)$ , the range of $\left\{ x \right\}$ is $\left[ 0,1 \right)$ , where $n$ is any integer. $\begin{align} &nbsp;&nbsp;& \Rightarrow \text{For any interval }\left[ n,n+1 \right),\text{ the range of sin}\left\{ x \right\}\text{ is }\left[ 0,\sin 1 \right). \\ &nbsp;& \text{We have the following intuition to obtain the above graph}\text{.} \\ &nbsp;& \therefore \text{The range of the function is }\left[ 0,\sin 1 \right). \\ \end{align}$