## Babynini one year ago Solve each equation in the interval [0,2pi) rounded to two decimal points a) (2cos(theta)-1)(sin(theta)-1)=0

1. Babynini

em I got 2cos(theta)sin(theta)-2cos(theta)-sin(theta)+1=0 yeah?

2. freckles

If a*b=0, then either a=0 or b=0 or both=0. This means you have: $2\cos(\theta)-1=0 \text{ or } \sin(\theta)-1=0$

3. freckles

isolate the trig function

4. freckles

$\cos(\theta)=\frac{1}{2} \text{ or } \sin(\theta)=1$

5. freckles

try to solve both of those equations

6. Babynini

cos(theta) = 1.047 sin(theta) = 1.57

7. freckles

oh you mean: $\theta \approx 1.05 \text{ you should also be able to obtain } \theta \approx 5.24 \text{ from } \cos(\theta)=\frac{1}{2}$ and $\theta \approx 1.57 \text{ from the other equation } \sin(\theta)=1$

8. Babynini

Yes, that is what I meant :) how did you get 5.24?

9. freckles

$\cos(\theta)=\cos(-\theta)=\cos(-\theta+2 \pi ) \\ \cos(1.05) \approx \cos(-1.05+2\pi) \approx \cos(-1.05+2(3.14)) \\ \cos(1.05) \approx \cos(-1.05+6.28)=\cos(5.23) \\ \text{ but honestly I used the unit circle }$

10. freckles
11. freckles

You see the points on the circle. The first number of every pair is the output for the cos(the angle there). The second number of every pair is the output for the sin(the angle there). I was looking for when the x-coordinate (the first number aka the cos number) was 1/2. I see this was happening at the pi/3 mark and the 5pi/3 mark.

12. freckles

And your question wanted these values rounded to the nearest hundredth.

13. freckles

$\frac{\pi}{3} \approx 1.05 \\ \frac{5 \pi}{3} \approx 5.24$

14. Babynini

ooo I see! that makes sense! haha why do we not want to do the othe pi/3 s?

15. freckles

at 2pi/3 and 4pi/3 you should see the cos number (the x-coordinate) is negative

16. freckles

1/2 is not a negative number

17. Babynini

at pi/3 it is not negative either. That was another question I have why do we make it -1.05 instead of adding 2pi to 1.05

18. freckles

cos is an even function $\cos(\theta)=\cos(-\theta)$

19. freckles

sin is an odd function $\sin(\theta)=-\sin(-\theta)$

20. Babynini

ooh yes.

21. freckles

and then inside you can add as many 2pi as you like because we just wind up at the same place

22. freckles

but I add one 2pi since that would give me another solution in [0,2pi)

23. freckles

Pretend we want to solve this: $\cos(\theta)=.23 , \theta \in [0,2\pi)$ This .23 is not on the unit circle. But using the unit circle and other identities helps me to solve this equation. First I can do arccos( ) on both sides to find one solution. $\theta=\arccos(.23)$ another solution would be given by: $-\theta+2\pi=\arccos(.23) \\ \theta=-\arccos(.23)+2\pi$ and if we wanted to approximate these solutions by rounding to the nearest hundredth we would have: $\theta \approx 4.94 , 1.34$

24. freckles

The second solution I used the fact that cos was even and had a period of 2pi.

25. Babynini

Hmm ok.

26. Babynini

http://openstudy.com/users/babynini#/updates/55823438e4b07028ea611c18 so for this one, how do I do that?

27. Babynini

Thank you for the help on this one, btw. :)

28. freckles

np