## anonymous one year ago Verify the identity. cos (x + pi/2) = -sin x

1. anonymous

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

2. Jhannybean

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

3. anonymous

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

4. Jhannybean

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

5. Jhannybean

think of the unit circle.

6. anonymous

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

7. Jhannybean

|dw:1440138751966:dw|

8. anonymous

so cos (1) sin (0) ?

9. Jhannybean

No.

10. Jhannybean

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

11. Jhannybean

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

12. anonymous

i see, i see.

13. 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?

14. anonymous

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

15. Jhannybean

Yes, and we sound what cos(90) and sin(90) were, so substitute those in. $\color{red}{\cos(90^\circ )} = \cos\left(\frac{\pi}{2}\right)=\color{red}{0}$ $\color{red}{\sin(90^\circ )} =\sin\left(\frac{\pi}{2}\right) = \color{red}{1}$

16. Jhannybean

found*

17. anonymous

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

18. Jhannybean

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

19. Jhannybean

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

20. anonymous

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

21. Jhannybean

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

22. 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}$ $\cos \left(x+\frac{\pi}{2}\right) = \cos (x)(\color{red}{0}) - \sin (x) (\color{red}{1})$

23. Jhannybean

Do you see what I mean now?

24. anonymous

yes i do!

25. Jhannybean

Can you simplify the rest from here?

26. anonymous

what would i do to simplify?

27. Jhannybean

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

28. anonymous

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

29. Jhannybean

Yay.

30. anonymous

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

31. 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.