Question

FIND THE MAXIMUM VALUE OF 2 SIN X - COS X.

Answer 1

For a question like this, you can combine the sine and cosine into a single function through a linear combination. Basically, anything of the form:
\[a\sin(x)+b\cos(x)=A\cos(x-C)\]Now, to make this happen, you can use the following formulas.\[A=\sqrt{a^2+b^2}\]\[C=\tan^{-1}\frac ab\]For the 'C' one, make sure that it is the sine coefficient over the cosine coefficient! Also make sure when doing something like this, the periods of the sine and cosine MUST be the same. Now, to find the maximum value, what you're worried about is the amplitude. So, you need to find the amplitude of the new, combined function. Knowing this, which of the formulas is important in this case?

Answer 2

however,2 sin x -cos x ...not +

Answer 3

The sign doesn't really matter. The 'b' on the cosine can be negative (adding a negative number is the same as subtracting).

Answer 4

Ok, we need to find the amplitude to figure out the maximum value. In my first comment, which of the variables (either A or C) is the amplitude? That's the formula we're going to have to use.

Answer 5

is the answer is 2 or \[\sqrt{5}\]

Answer 6

It is one of those two, but can you show how you got to those?

Answer 7

THE TWO IS THE AMPLITUDE

Answer 8

SO I THOUGHT TTWO WOULD BE THE MAXIMUM VALUE WHEN U DRAW THE GRAPH

Answer 9

let f(x)=2 sin x-cos x
f'(x)=2 cos x+sin x
f'(x)=0 gives
2 cos x+sin x=0
sinx=-2cosx
tan x=-2
f"(x)=-2 sin x+cos x