## 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. anonymous

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

3. anonymous

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

4. anonymous

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

5. anonymous

think of the unit circle.

6. anonymous

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

7. anonymous

|dw:1440138751966:dw|

8. anonymous

so cos (1) sin (0) ?

9. anonymous

No.

10. anonymous

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

11. anonymous

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

12. anonymous

i see, i see.

13. anonymous

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. anonymous

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. anonymous

found*

17. anonymous

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

18. anonymous

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

19. anonymous

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. anonymous

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

22. anonymous

$\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. anonymous

Do you see what I mean now?

24. anonymous

yes i do!

25. anonymous

Can you simplify the rest from here?

26. anonymous

what would i do to simplify?

27. anonymous

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. anonymous

Yay.

30. anonymous

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

31. anonymous

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.