Homework Help: Finding critical numbers of trig function

1. May 2, 2010

TsAmE

1. The problem statement, all variables and given/known data

Find the critical numbers of the function:

f(θ) = 2cosθ + sin^(2) θ

2. Relevant equations

None

3. The attempt at a solution

I differentiated the equation and got -2sinθ(1 - cosθ) and found the critical values to be θ = 0 degrees + 2pie * n but the correct answer was npie. Why is this?

2. May 2, 2010

justsof

When is -2sinθ(1 - cosθ) equal to zero according to you? (check again I mean)

3. May 8, 2010

TsAmE

cosθ = 1 when x = 0 + 2npie, n E Z
-2sinθ = 0 when x = 0 + 2npie, n E Z

4. May 8, 2010

Staff: Mentor

Edit: Corrected an error I made.
cosθ = 1 for x = n*2pi
sinθ = 0 for x = ..., -2pi, - pi, 0, pi, 2pi, 3pi, ...

Last edited: May 9, 2010
5. May 9, 2010

justsof

No Mark! cosθ = 1 for θ = n*2pi, so you are correct TsAmE.
but you indeed made an error with the sinus:

sin(x)= 0 for x = n*pi

6. May 9, 2010

TsAmE

Why does sinx have the same value every pie (180 degrees)? Doesnt it only repeat every 2pie since it is its period?

7. May 9, 2010

Staff: Mentor

Right. I don't know what I was thinking. I have edited my earlier reply.

8. May 9, 2010

Staff: Mentor

The graphs of y = cosx and y = sinx are periodic with period 2pi, but both cross the horizontal axis at multiples of pi. But that doesn't mean that the period of each is pi. For periodicity, f(x + p) = f(x) for all x, not just a select few values.

9. May 11, 2010

TsAmE

What do you mean by f(x + p) = f(x) and what is your f(x). What if my critical number was t = pie/6 OR t = -pie/2, would the critical values be at pie/6 + npie OR t = -pie/2 + npie?

Last edited: May 11, 2010
10. May 11, 2010

Staff: Mentor

That's the definition of periodicity. If a function f satisfies f(x + p) = f(x) for all x, f is periodic with period p.

The cosine and sine functions are periodic with period 2pi. BTW, pie is something you eat. Pi is the name of the Greek letter $\pi$. If you want to appear intelligent, don't write pie when you mean pi.

If your critical number was t = pi/6 for the cosine or sine function, then pi/6 + n(2pi) would also be a critical number.

11. May 11, 2010

TsAmE

Lol I didnt even notice that I wrote pie. So the only time that you add "n(pi)" to the period is when sinx or cosx = 0, and for any other values you would instead add "2n(pi)"?

12. May 11, 2010

Staff: Mentor

Right.

13. May 12, 2010

TsAmE

Are there any other special cases where you wouldnt say "+ 2n(pi)"?

14. May 12, 2010

Staff: Mentor

No, not for the sine and cosine functions.