Determine the limits of the range of y given by y=cos2x-4sinx .

In other words, we'll have to identify the maximum and
minimum values for y.

We'll re-write the expression of y,
using just sine function.

We'll apply the double angle
identity for the first term.

cos 2x = 1 - 2(sin
x)^2

y = 1 - 2(sin x)^2 - 4 sin
x

We'll consider the maximum value for sin x =
1:

y = 1 - 2 - 4

y = -6 +
1

y = -5

Now, we'll consider
the minimum value for sin x = -1.

y = 1 - 2 +
4

y = 3

So, for a maximum
value for sin x, we'll get a minimum value for y, namely y = -5. For a minimum value of
sin x, we'll get a maximum of y = 3.

The
range of values of y, if y = cos2x-4sinx, is represented by the closed interval [-5 ;
3].