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This is the second of our many quizzes on Graphing trigonometric functions. Click on this to access more such competitive quizzes on Trigonometry.

Graphing trigonometric functions Quiz 2

mathematics

What is the time period of the function given below <img src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/afc54dd0-2c02-4deb-ae9c-8051ca6d468c2167221428648088539.png"/>

 Try to find the length of the interval in which the graph is repeating itself, then calculate the time period for the function.

The period of the graph is the value of $'x'$ for which the function is repeating. Here there are two branches of the graph that are repeating. From the graph, a full upward branch and half branch of the repeating which is ${{\left( \dfrac{3}{4} \right)}^{th}}$ part of the graph is given in the domain $\left[ 0,\dfrac{3\pi }{2} \right]$ . So, if the time period is $T$then the time period of ${{\left( \dfrac{3}{4} \right)}^{th}}$ part of graph is given as, $\Rightarrow \dfrac{3T}{4}=\dfrac{3\pi }{2}-0 \\ \Rightarrow T=2\pi $

 Find the number of solutions of the equation $\sin \left( 2x \right)=\tan x$ in the interval $\left[ 0,2\pi  \right]$.

 Make the graph for $y=\sin \left( 2x \right)$ and $y=\tan x$ . To make it, take the time period of the function as $\pi $ . Shape is similar to the graph of $y=\sin x$ . Then calculate the number of points where these two curves are intersecting or touching.

<img src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/c060be62-35bb-495d-a0c9-5ad89801ebe02211297248935392161.png"/> To make the graph for $y=\sin \left( 2x \right)$ , take the time period of the function as $\pi $ . Shape is similar to the graph of $y=\sin x$ . To make the graph for function $y=\tan x$ , use the time period of the function as $\pi $ and the range as $\left( -\infty ,\infty \right)$. The function $y=\tan x$ is increasing function. There are 7 points of intersection between the two curves. So, the number of solutions for the given equation is 7.

 Find the range of the function $\max \{\sin x,\cos x\}$ in the interval $\left[ 0,2\pi  \right]$.

 Draw the graph for $y=\cos x$ and $y=\sin x$ . Then find where one out of two is maximum, then find the range.

<img src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/8aaad598-ed96-4395-9644-37a7ed070dea4354939733116547993.png"/> <img src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/7e1ea2b2-d1e9-4401-89c6-97a91ebc913b1716861101886250101.png"/> To make the graph for $\max \{\sin x,\cos x\}$ , 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. So from the graph range of the function is $\left[ -\dfrac{1}{\sqrt{2}},1 \right]$.

 Find the number of points of the non-differentiability of the  function $\max \{\sin x,\cos x\}$ in the interval $\left( 0,2\pi  \right)$.

 Draw the graph for $y=\cos x$ and $y=\sin x$. Then find where one out of two is maximum. From the graph obtained, find the points of non-differentiability.

<img src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/0f369e4e-2287-49bd-972e-8f9563c370c16539461751660829337.png"/> <img src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/e0873184-2457-4663-b4a1-ebfee525e85f2373682917551991337.png"/> To make the graph for $\max \{\sin x,\cos x\}$ , 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. So the obtained graph has a sharp corner point at $x=\dfrac{\pi }{4}\text{ and }x=\dfrac{5\pi }{4}$ . So, there are two points of non-differentiability.

 Find the time period of the function $\left| \max \{\sin x,\cos x\} \right|$.

 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 the abscissa keeping the graph on the positive side of the ordinate as it is. Then analyse the graph according to the question.

<img src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/edf808eb-53b3-4702-a4b5-18970c3ed6752773058095872555740.png"/> <img src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/4228ce94-b6b2-42f1-85aa-725387937e0f8267469987642450757.png"/> To make the graph for $\max \{\sin x,\cos x\}$ , 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 $\left| \max \{\sin x,\cos x\} \right|$ , 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/d1a150b0-b85f-4cdb-bcaa-bb73eccd6137946867407535029561.png"/> So, the time period of the above graph $=2\times \left( \dfrac{5\pi }{4}-\dfrac{\pi }{4} \right)=2\pi $.