Maxima and Minima Trig Problem

Hello,

I'm having difficulty in understanding where the book has got these angles from:

Question: Find the maximum and minimum values for

Now, in my book the answer is 5 at for a maximum value, and -5 at for a minimum value. However, they have subtracted from in order to obtain these pair of angles? I've drawn the function, and these values is where . As a matter of fact, the maximum value first occurs at and the minimum value at .

I've looked into this a bit deeper, and the only way a maximum or minimum value can occur at these computed angles - is if the first derivative is a negative cosine function.

What have I done wrong?

Thank you for your attention.

Re: Maxima and Minima Trig Problem

first off, the problem states the interval of interest is ... you need to start thinking radians.

for , and will have to have opposite signs ... which tells you will have to be in quadrants II or IV.

, a quad II angle

, a quad IV angle

to finish, use the second derivative test for the two angles to determine max/min.

Re: Maxima and Minima Trig Problem

Quote:

for

,

and

will have to have opposite signs ... which tells you

will have to be in quadrants II or IV.

, a quad II angle

, a quad IV angle

to finish, use the second derivative test for the two angles to determine max/min.

I don't understand. For why do we have to give sin and cos opposite signs to equal zero?

If sin and cos are both made positive does this not make ?

Further, why have you added and to ?

Re: Maxima and Minima Trig Problem

Quote:

Originally Posted by

**astartleddeer** I don't understand. For

why do we have to give sin and cos opposite signs to equal zero?

3(something) + 4(something else) = 0

what does that tell you about the signs of (something) and (something else) ?

Quote:

If sin and cos are both made positive does this not make

?

if sine and cosine have the same sign, then tangent is positive ... the solutions for y' = 0 is where tangent is negative.

Quote:

Further, why have you added

and

to

?

you need to research the range of the arctangent function.

Re: Maxima and Minima Trig Problem

Quote:

Originally Posted by

**skeeter** 3(something) + 4(something else) = 0

what does that tell you about the signs of (something) and (something else) ?

Then only one of them can be negative?

Quote:

If sine and cosine have the same sign, then tangent is positive ... the solutions for y' = 0 is where tangent is negative.

Oops. I was meant to say, if sin and cos are both negative then the tangent is positive in the third quadrant?

Quote:

you need to research the range of the arctangent function.

Oh my goodness! I had mistaken for ! Yes, I'm with you now. If the tangent is negative, it is or . Second and fourth quadrant respectively.

Re: Maxima and Minima Trig Problem

Ok let's do this.

and

is a maximum for

is a minimum for

Therefore, a minimum value occurs at and a maximum value occurs at

Re: Maxima and Minima Trig Problem

Re: Maxima and Minima Trig Problem

I indicated radians with a small c for circular measure (Happy)

Ok, thank you for your time Skeeter. Done and dusted (Clapping)