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Trigonometry

 

 

OldTimer  Aug 24, 2018
 #1
avatar+93289 
+4

\(\text{Sketch the graph  }\quad y= \frac{\pi}{2}\; sin^2x  \quad  \text{and use it to solve the equation  }\quad2x=\pi\;sin^2x\)

 

https://www.desmos.com/calculator/iyzyjfqbzc

 

 

solutions are x = 0, pi/4 and  pi/2 radians

Melody  Aug 24, 2018
 #2
avatar+93289 
+4

\(\text{Sketch the graph }\quad y= \frac{\pi}{2}\; sin^2x \quad \text{and use it to solve the equation }\quad2x=\pi\;sin^2x\)

 

 

I have been asked how to plot the original graph.

 

I would 'build' this graph.  Starting with y=sinx

This has a period of 2pi and a amplitude of 1

 

x (radians)0pi/4pi/2pi3pi/22pi
sin x0sqrt2/2=0.7110-10
(sinx)^200.51010
pi/2 (sinx)^20pi/4pi/20pi/20
       
       

 

So you can graph this.

 

\( \text{Not that }\quad2x=\pi\;sin^2x \quad \text{is the same as } x=\frac{\pi}{2}\;sin^2x\)

 

So you want the intersection of     \(y=\frac{\pi}{2}\;sin^2x\)    with   \(y=x\)

 

Draw the sine graph and the graph of y=x and it is very easy to see where they intersect.

 

I hope that help  laugh

Melody  Aug 24, 2018
 #3
avatar+93289 
+5

I have been asked if I can try and explain a little more.

 

\(\text{Sketch the graph }\quad y= \frac{\pi}{2}\; sin^2x \quad \text{and use it to solve the equation }\quad2x=\pi\;sin^2x\)

 

 

\(2x=\pi\;sin^2x\\ \text{rearranging we get}\\ x=\frac{\pi}{2}sin^2x\)

 

 

\(\text{So we have}\quad y= \frac{\pi}{2}\; sin^2x \quad\\ \text{and we want to use this graph to determine when } \quad x= \frac{\pi}{2}\; sin^2x \\ \text{but we have the graph of } \quad y= \frac{\pi}{2}\; sin^2x \quad \\ \text{So we want to know when x=y on this graph.}\\ \text{i.e. }\quad y=x=\frac{\pi}{2}\; sin^2x\\ \text{So if we find the intersection of } \qquad y=\frac{\pi}{2}\; sin^2x \qquad and \qquad y=x\\ \text{That will be all the solutions of }\qquad x= \frac{\pi}{2}\; sin^2x \)

 


\(\text{The intersection happens when }\quad    x= 0, \;\frac{\pi}{4} \;\;and  \;\frac{\pi}{2}         \\ \text{These are the only solutions to  }  x=\frac{\pi}{2}sin^2x\\\)

 

 

I hope that helps more   wink

Melody  Aug 24, 2018
edited by Melody  Aug 24, 2018
edited by Melody  Aug 24, 2018
edited by Melody  Aug 24, 2018
 #4
avatar+93289 
+2

Hi Oldtimer,

If x is in degrees then this doesn't work the same. The only solution would be  x=0

 

For any higher mathematics it should always be assumed that angles are in radians.

If for some reason, degrees are required then the degrees sign should be displayed.

Of course for many questions it does not matter but here it does matter.

 

Here are the graphs if degrees are used.

 

Melody  Aug 25, 2018

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