Question

Verify the identity.

cos (x + pi/2) = -sin x

Answer 1

\[\cos \left( x + \frac{ \pi }{ 2 } \right) = - \sin x\]

Answer 2

\[\color{red}{\cos(a+b) = \cos(a)\cos(b)-\sin(a)\sin(b)}\]

Answer 3

cos ( x + pi/2 ) = cos (x) cos (pi/2) - sin (x) sin (pi/2) ?

Answer 4

Good. now at \(\dfrac{\pi}{2}\) what is the value of \(\cos(\theta)\) and \(\sin(\theta)\)?

Answer 5

think of the unit circle.

Answer 6

cos (0) sin (1) ? or is it the other way around?

Answer 7

so cos (1) sin (0) ?

Answer 8

No.

Answer 9

\[\cos(90^\circ ) = \cos\left(\frac{\pi}{2}\right)=0\]\[\sin(90^\circ ) =\sin\left(\frac{\pi}{2}\right) = 1\]

Answer 10

sine functions represent y-values, and cosine functions represent x-values. Remember that.

Answer 11

i see, i see.

Answer 12

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?

Answer 13

cos ( x + pi/2 ) = cos (x) cos (90) - sin x sin (90) ?

Answer 14

Yes, and we sound what cos(90) and sin(90) were, so substitute those in.

\[\color{red}{\cos(90^\circ )} = \cos\left(\frac{\pi}{2}\right)=\color{red}{0}\]
\[\color{red}{\sin(90^\circ )} =\sin\left(\frac{\pi}{2}\right) = \color{red}{1}\]

Answer 15

found*

Answer 16

cos ( x + pi/2) = cos (x) cos (0) - sin (x) sin(1)?

Answer 17

No, replace the values, 0 and 1, in the appropriate places.

Answer 18

No sin and cos needed. They EQUAL eachother, therefore sin(90) and cos(90) can be REPLACED by 0 and 1.

Answer 19

cos (x + pi/2) = 0 - 1

Answer 20

I don't think you understand what im saying...

Answer 21

\[\color{red}{\cos(90^\circ )} = \cos\left(\frac{\pi}{2}\right)=\color{red}{0}\]\[\color{red}{\sin(90^\circ )} =\sin\left(\frac{\pi}{2}\right) = \color{red}{1}\] \[\cos \left(x+\frac{\pi}{2}\right) = \cos (x)(\color{red}{0}) - \sin (x) (\color{red}{1})\]

Answer 22

Do you see what I mean now?

Answer 23

yes i do!

Answer 24

Can you simplify the rest from here?

Answer 25

what would i do to simplify?

Answer 26

Think about what anything multiplied by 0 is, and what happens when you multiply a number by 1.

Answer 27

cos ( x + pi/2 ) = - sin x
i see.

Answer 28

Yay.

Answer 29

sorry if i gave you a hard time, but thank you so much for your help!

Answer 30

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.