Free Study Material

NCERT Solutions

NCERT Books

Revision Notes

Practice Problems

This is one of our many quizzes on Graphing sin, cos, tan functions. Click on this to access more such competitive quizzes on Trigonometry.

Graphing sin, cos, tan functions Quiz 2

mathematics

 Find the equation for the following graph: $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/eb908147-15fb-4f32-8e66-8adbbd13a5b98773802150627130116.png"/>

 Observe the graph and find whether it is stretched or not and then find , by how many units , the graph is shifted to obtain the equation of the given graph.

  The general form of a sine graph is $y=a\sin \theta +q$ $ \\ $ In the given graph, the graph is shifted upwards by $1$ unit $ \\ $ Also the graph is not stretched. Hence, the equation of given graph is $y=\sin \theta +1$

 The graph below shows a trigonometric equation of the following form : $y=a\cos \theta +q$. Two points are shown on the graph : Point A at $\left( {{90}^{\circ }},0 \right)$ and Point B at $\left( {{180}^{\circ }},-0.5 \right)$. Calculate the values of $a$ and $q$. $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/a29571f3-1dcd-42ca-8bbc-80109e166ea93181875525352575006.png"/>

 First find the value of $a$ by using point B and then find the value of $q$ by observing whether the graph is shifted or not.

  The y-value of point $B$ is $-0.5$ while the y-value of $A$ is $0$ $ \\ $ Therefore , amplitude $a=0-\left( -0.5 \right)=0.5=\dfrac{1}{2}$ $ \\ $ $q$ shifts the graph up or down. Point $A$ is at $\left( {{90}^{\circ }},0 \right)$. $ \\ $ For an unshifted cosine graph with the same $a$ value $\left( \text{i}\text{.e }\dfrac{1}{2}\cos \theta  \right)$ point $B$ would be at $\left( {{0}^{\circ }},0 \right)$ $ \\ $ For this graph, point has not been shifted. Therefore $q=0$ $ \\ $ The complete equation for the graph shown in this question is $y=\dfrac{1}{2}\cos \theta$ $ \\ \therefore a=\dfrac{1}{2}\text{ and }q=0$

 Shown the following graph of the following form : $y=a\sin \theta +q$where Point A is at $\left( {{180}^{\circ }},1.5 \right)$  and  Point B is at $\left( {{90}^{\circ }},3 \right)$.Find the values of $a$ and $q$.$ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/df8b541b-20e8-4a2c-ba41-fc2f96f76bb54170359735951397788.png"/>

 From the graph, first find the value of $q$ and then substitute point B in the given equation to find the value of $a$.

Note that $q$ shifts the graph up or down. Point $A$ is at $( {{180}^{\circ }},1.5)$. For an unshifted tangent graph, point $A$ would be at $(180^{\circ },0)$. This point is shifted by $\dfrac{3}{2}$ units. Hence $q=\dfrac{3}{2}$.  $ \\ $ Substituting Point $B$ in the given equation, $ \\ y=a\sin \theta +\dfrac{3}{2} \\  \Rightarrow 3=a\sin {{90}^{\circ }}+\dfrac{3}{2} \\  \Rightarrow \dfrac{3}{2}=a\left( 1 \right) \\  \Rightarrow a=\dfrac{3}{2} \\ $ Hence, $a=\dfrac{3}{2}\text{and }q=\dfrac{3}{2}$

 Find the number of solutions for $\tan \left( \dfrac{3}{2}x \right)=2$, where $x\in \left[ -\pi ,\pi  \right]$

 First draw the graph of $y=\tan \left( \dfrac{3}{2}x \right)$ and $y=2$, then find the points of intersection between $-\pi $ and $\pi $.

<sub></sub><img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/6aff2914-f723-4eda-809a-337c45df9b598815797076902149936.png"/>$ \\ $ Given $\tan \left( \dfrac{3}{2}x \right)=2$ $ \\ $ Consider $y=\tan \left( \dfrac{3}{2}x \right)$ and $y=2$ $ \\ $ Draw the graph of $y=\tan \left( \dfrac{3}{2}x \right)\text{ and }y=2$ $ \\ $ As $x\in \left[ -\pi ,\pi  \right]$ there are only three points of intersection of $y=\tan \left( \dfrac{3}{2}x \right)$ and $y=2$ $ \\ $ Hence the number of solutions is $3$.

 If the number of solutions of $5\cos \dfrac{1}{2}\left( x+\pi  \right)=x$,$x\in \left[ -4\pi ,4\pi  \right]$  be $\lambda $. Then find the value of $\left[ \dfrac{\lambda }{3} \right]$ $($where $[.]$ denotes the greatest integer function$)$

 First find the value of $\lambda $ by drawing the graphs of $y=5\cos \dfrac{1}{2}\left( x+\pi  \right)$ and $y=x$ and then find the value of $\left[ \dfrac{\lambda }{3} \right]$.

 <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/40d84d32-8ec7-4a89-93a8-32cf7536a1d45047062910638545237.png"/> $ \\ $ Given $5\cos \dfrac{1}{2}\left( x+\pi  \right)=x$ $ \\ $ Draw the graph of $y=5\cos \dfrac{1}{2}\left( x+\pi  \right)$ and $y=x$ $ \\ $ As $x\in \left[ -4\pi ,4\pi  \right]$, there is only one point of intersection of  $y=5\cos \dfrac{1}{2}\left( x+\pi  \right)$ and $y=x$ $ \\ $ Hence the number of solutions is $1$. $ \\ \Rightarrow \lambda =1 \\  \therefore \left[ \dfrac{\lambda }{3} \right]=0$