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.