Finding critical numbers of trig function

[TsAmE]

Homework Statement

Find the critical numbers of the function:

f(θ) = 2cosθ + sin^(2) θ

Homework Equations

None

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?

 

[justsof]

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

 

[TsAmE]

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

 

[Mark44]

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

 

[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

 

[TsAmE]

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

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

 

[Mark44]

No Mark! cosθ = 1 for θ = n*2pi, so you are correct TsAmE.

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

but you indeed made an error with the sinus:

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

 

[Mark44]

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

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.

 

[TsAmE]

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.

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?

 

[Mark44]

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.

 

[TsAmE]

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.

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)"?

 

[Mark44]

Right.

 

[TsAmE]

Are there any other special cases where you wouldn't say "+ 2n(pi)"?

 

[Mark44]

No, not for the sine and cosine functions.