Find the range of values of the trigonometric function y = -2sinx.

The range of values ​​of the sinx function
Some properties of the trigonometric sinx function.

The function is a periodic and odd function. Function period: 2π = 360 °. For some argument values:

x0 = 0 = 0 °;
x1 = π / 6 = 30 °;
x2 = π / 4 = 45 °;
x3 = π / 3 = 60 °;
x4 = π / 2 = 90 °;
the function value has a simple algebraic form:

sin (0) = 0;
sin (π / 6) = 1/2;
sin (π / 4) = √2 / 2;
sin (π / 3) = √3 / 2;
sin (π / 2) = 1.
To remember these values, it is enough to use the following pattern – the sinx value for the i-th angle can be calculated with the same expression:

sin (xi) = √i / 2.

Really:

sin (x0) = √0 / 2 = 0/0 = 0;
sin (x1) = √1 / 2 = 1/2;
sin (x2) = √2 / 2;
sin (x3) = √3 / 2;
sin (x4) = √4 / 2 = 2/2 = 1.
Thus, in the range of values ​​of the argument [0; π / 2] sinx increases from 0 to 1, therefore, the range of values ​​of the function for this interval: [0; 1].

In the second quarter, sinx decreases from 1 to 0, and the range of values ​​is: [0; 1].

In the third quarter, sinx decreases from 0 to -1, range of values: [-1; 0].

In the fourth quarter sinx again increases from -1 to 0, range of values: [-1; 0].

As a result, we get the range of values ​​for sinx: [-1; 1]. This means that for any value of x, the double inequality is true:

-1 ≤ sinx ≤ 1. (1)

Range of values ​​of the original function
We multiply all parts of inequality (1) by -2, changing the signs of the inequality and bringing it to the usual form:

-2 * (-1) ≥ -2 * sinx ≥ -2 * 1;

2 ≥ -2sinx ≥ -2;

-2 ≤ -2sinx ≤ 2.

From this inequality it follows that the range of values ​​of the function -2sinx: [-2; 2].

Answer: [-2; 2].