The largest and smallest value of the expression 3 sin x-1 and 5-4sin x

The sine function of the argument takes values that are in the range [-1; 1]. We write the range of values of the sine function of the argument in the form of a double inequality:

-1 <= sin x <= 1;

Multiply all parts of the inequality by 3:

-3 <= 3 * sin x <= 3;

Subtract one from all parts of the inequality:

-4 <= 3 * sin x <= 2.

The smallest value is -4, the largest is 2.

2) -1 <= sin x <= 1;

Multiply all parts by -4:

-4 <= -4 * sin x <= 4;

Let’s add five to these parts:

1 <= 5 – 4 * sin x <= 9.

The smallest value is 1, the largest is 9.