Prove that (2√3+3)sinx+2√3cosx lies between -(2√3+15) and (2√3+15)

To prove that the expression (2√3+3)sin(x) + 2√3cos(x) lies between -(2√3+15) and (2√3+15), we need to find the minimum and maximum values of this expression.

Let's start by simplifying the expression:

(2√3+3)sin(x) + 2√3cos(x)
= 2√3sin(x) + 3sin(x) + 2√3cos(x)
= (2√3sin(x) + 2√3cos(x)) + 3sin(x)

Now, we can rewrite this expression using the trigonometric identity: sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

= 2√3(sin(x)cos(π/3) + cos(x)sin(π/3)) + 3sin(x)

By applying the identity, we get:

= 2√3sin(x + π/3) + 3sin(x)

Now, let's analyze the individual terms:

The range of sin(x) is [-1, 1]. So, the minimum value of 3sin(x) is -3, and the maximum value is 3.

Similarly, the range of sin(x + π/3) is also [-1, 1]. So, the minimum value of 2√3sin(x + π/3) is -2√3, and the maximum value is 2√3.

Now, let's find the minimum and maximum values of the entire expression:

Minimum value: -2√3 - 3 = -2√3 - 3
Maximum value: 2√3 + 3 = 2√3 + 3

Therefore, we can conclude that the expression (2√3+3)sin(x) + 2√3cos(x) lies between -(2√3+15) and (2√3+15), as the minimum value is greater than -(2√3+15) and the maximum value is less than (2√3+15).

To prove that (2√3+3)sinx+2√3cosx lies between -(2√3+15) and (2√3+15), we need to show that the given expression is greater than -(2√3+15) and less than (2√3+15) for all values of x.

Let's start by simplifying the given expression:

(2√3+3)sinx + 2√3cosx

Using the formula for the sum of angles for sine and cosine, we can rewrite the expression as:

√3(sin(x+30°) + cos(x))

Now, let's examine the individual terms:

sin(x+30°) lies between -1 and 1 for all values of x.
cos(x) lies between -1 and 1 for all values of x.

Therefore, the absolute value of the given expression is less than or equal to (√3 + 1) + 2√3 = 3√3 + 1, for all values of x.

Now, let's compare this to the bounds given:

-(2√3+15) < 0, and (2√3+15) > 0.

Since 3√3 + 1 > 0, the given expression is indeed greater than -(2√3+15) and less than (2√3+15) for all values of x.

Therefore, we have proved that (2√3+3)sinx+2√3cosx lies between -(2√3+15) and (2√3+15).

-(√12+9+12√3+12)<(2√3+3)sinx+2√3cosx<(√12+9+12√3+12)