anonymous
  • anonymous
Verify the identity. cos (x + pi/2) = -sin x
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
\[\cos \left( x + \frac{ \pi }{ 2 } \right) = - \sin x\]
Jhannybean
  • Jhannybean
\[\color{red}{\cos(a+b) = \cos(a)\cos(b)-\sin(a)\sin(b)}\]
anonymous
  • anonymous
cos ( x + pi/2 ) = cos (x) cos (pi/2) - sin (x) sin (pi/2) ?

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Jhannybean
  • Jhannybean
Good. now at \(\dfrac{\pi}{2}\) what is the value of \(\cos(\theta)\) and \(\sin(\theta)\)?
Jhannybean
  • Jhannybean
think of the unit circle.
anonymous
  • anonymous
cos (0) sin (1) ? or is it the other way around?
Jhannybean
  • Jhannybean
|dw:1440138751966:dw|
anonymous
  • anonymous
so cos (1) sin (0) ?
Jhannybean
  • Jhannybean
No.
Jhannybean
  • Jhannybean
\[\cos(90^\circ ) = \cos\left(\frac{\pi}{2}\right)=0\]\[\sin(90^\circ ) =\sin\left(\frac{\pi}{2}\right) = 1\]
Jhannybean
  • Jhannybean
sine functions represent y-values, and cosine functions represent x-values. Remember that.
anonymous
  • anonymous
i see, i see.
Jhannybean
  • Jhannybean
going back to our function , \(\color{red}{\cos(x+\frac{\pi}{2} ) = \cos(x)\cos(\frac{\pi}{2})-\sin(x)\sin(\frac{\pi}{2})}\) can you replace the newfound values and solve for it?
anonymous
  • anonymous
cos ( x + pi/2 ) = cos (x) cos (90) - sin x sin (90) ?
Jhannybean
  • Jhannybean
Yes, and we sound what cos(90) and sin(90) were, so substitute those in. \[\color{red}{\cos(90^\circ )} = \cos\left(\frac{\pi}{2}\right)=\color{red}{0}\] \[\color{red}{\sin(90^\circ )} =\sin\left(\frac{\pi}{2}\right) = \color{red}{1}\]
Jhannybean
  • Jhannybean
found*
anonymous
  • anonymous
cos ( x + pi/2) = cos (x) cos (0) - sin (x) sin(1)?
Jhannybean
  • Jhannybean
No, replace the values, 0 and 1, in the appropriate places.
Jhannybean
  • Jhannybean
No sin and cos needed. They EQUAL eachother, therefore sin(90) and cos(90) can be REPLACED by 0 and 1.
anonymous
  • anonymous
cos (x + pi/2) = 0 - 1
Jhannybean
  • Jhannybean
I don't think you understand what im saying...
Jhannybean
  • Jhannybean
\[\color{red}{\cos(90^\circ )} = \cos\left(\frac{\pi}{2}\right)=\color{red}{0}\]\[\color{red}{\sin(90^\circ )} =\sin\left(\frac{\pi}{2}\right) = \color{red}{1}\] \[\cos \left(x+\frac{\pi}{2}\right) = \cos (x)(\color{red}{0}) - \sin (x) (\color{red}{1})\]
Jhannybean
  • Jhannybean
Do you see what I mean now?
anonymous
  • anonymous
yes i do!
Jhannybean
  • Jhannybean
Can you simplify the rest from here?
anonymous
  • anonymous
what would i do to simplify?
Jhannybean
  • Jhannybean
Think about what anything multiplied by 0 is, and what happens when you multiply a number by 1.
anonymous
  • anonymous
cos ( x + pi/2 ) = - sin x i see.
Jhannybean
  • Jhannybean
Yay.
anonymous
  • anonymous
sorry if i gave you a hard time, but thank you so much for your help!
Jhannybean
  • Jhannybean
That's ok. As long as you understand the method used so you can ask new questions instead of ones where you're applying the same method over and over again.

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