harmonic identities question

Pretty much agree with your working. I guess the confusion is where youve written
min <= cos <= max
then seemingly multiplied by -1 and roughly argued the min is now the max. Youre dealing with -cos to begin with and the min and max of -cos are -1 and 1 ....

The maximum of the expression must be when cos = -1.

but i have multiplied with a negative number -root109 wont that change the inequality?

Youre honestly overthinking this, and the obvious answer is the correct one I hope you can see. Your answer has max < min which has to be wrong.

But to try and work through your reasoning
-1 (min) <= cos <= 1 (max)
and multiplying by -1 gives
1 >= -cos >= -1
The maximum of -cos corresponds to the negative minimum of cos, which is as it should be. You seem to have simply exchanged min and max without considering the sign. However, it should be common sense that for
2-3cos()
the max is 5 corresponding to cos=-1 and the min is -1 corresponding to cos=1

stil not really getting it, sorry

Which part? Do you agree that the min and max of
2 - 3cos()
are -1 and 5 respectively and they correspond to cos=1 and -1.

If so, try and explain your reasoning clearly for -cos() (forget about the 11 and the sqrt(109)) and how the min/max relate to the min/max of cos.

i think the inequality that my teacher told me is confusing me
i think i got it now so to be clear cos=1 max and cos=-1 happens only when i multiply with a negative number right?
if sqrt109 was positive max would still be cos=-1 and min still be cos=-1?

Fully agree with the bold. Its overkill for this question and seeing its motorhead monday (again ...)


cos takes values between -1 and 1, so does -cos. The min of both cos and -cos is -1 and the max of both cos and -cos is 1 (fairly obviously). So for simple linear expressions like
2 - 3cos()
(or your expression) the min is -1 and the max is 5. The - sign in front of the 3 doesnt affect the min/max values, for instance the max occurs when cos is -1 and you have 2+3.

so what i asked you
i think i got it now so to be clear cos=1 max and cos=-1 happens only when i multiply with a negative number right?

if sqrt109 was positive max would still be cos=-1 and min still be cos=-1?
is right?

I think so, the max of
+/-sqrt(109)cos(..)
is sqrt(109) irrespective of the sign and it occurs when cos(...)=1 if its +cos() and it occurs when cos(...)=-1 if its -cos(...). This question part only asks about the max value, so sqrt(109), but in part c) youd use the knowledge that this occurred when cos(...)=-1.

so in my mind in these king of questions when i see a multiplication of a negative number i should think that max cos=-1 and min cos=1?
same with sine with the question had sine?

Yes. If you just substitute the numbers in so for
2 - 3cos()
the max will be 2+3 which is when cos=-1 and the min is 2-3 which is when cos=1. As in #5, dont overthink it.

when multiply with possitive then the max is cos=1 and min cos=-1?
same with sine?

Yes. Its good to confirm with someone, but it should be obvious by just trying to sub the values +/-1 for the appropriate +/-trig term.

Yes. Its good to confirm with someone, but it should be obvious by just trying to sub the values +/-1 for the appropriate +/-trig term.

okay so i am right when negative i should do what we said and when possitive max. cos=1 and for min cos=-1?