Question

number of integers in the range of function

3sinx + 2 (root 3) cosx+3

Answer 1

3^2 + (2√3)^2 = 9+12 = 21

so, if cosy = 3/√21, you have
3sinx + 2√3 cosx = √21 (3/√21 sinx + 2√3/√21 cosx)
= √21 sin(x+y)
so the range of
3sinx + 2√3 cosx + 3 is [3-√21 , 3+√21] ≈ [-1.58,7.58]
How many integers in that?

Answer 2

9 integers

@oobleck am i right or wrong

Answer 3

surely you don't need a confirmation of that result.

But yes, you are right: -1, 0, 1, 2, 3, 4, 5, 6, 7

Answer 4

[AI-generated answer — Explain Bot]

To determine the number of integers in the range of the function f(x) = 3sinx + 2√3cosx + 3, we can analyze the properties of the sine and cosine functions.

The range of the sine function is [-1, 1], while the range of the cosine function is also [-1, 1]. This implies that the maximum range of the sum of the sine and cosine functions, 3sinx + 2√3cosx, will be [-3√3, 3√3].

Adding a constant term of 3 to this range will shift the range upwards by 3 units. Therefore, the range of the given function f(x) = 3sinx + 2√3cosx + 3 is [3 - 3√3, 3 + 3√3].

To determine the number of integers in this range, we need to find the number of integers between the two values 3 - 3√3 and 3 + 3√3.

Since the square root of 3 (√3) is an irrational number, the expression 3 - 3√3 is also an irrational number. On the other hand, the expression 3 + 3√3 is also an irrational number.

Hence, there are no integers in the given range.