Question

If (√3 sin θ = cos θ), find the value of (3 cos² θ + 2 cos θ/3 cos θ + 2).

Answer 1

The value of the expression (3 cos² θ + 2 cos θ/3 cos θ + 2) when (√3 sin θ = cos θ) is 14/3.

Step-by-step explanation:

To find the value of the expression (3 cos² θ + 2 cos θ/3 cos θ + 2) when (√3 sin θ = cos θ), we can substitute the value of cos θ from the given equation. Since (√3 sin θ = cos θ), we know that sin θ = (√3/3).

Now, let's substitute this value into the expression:

3 cos² θ + 2 cos θ/3 cos θ + 2 = 3(1 - sin² θ) + 2 cos θ/3 cos θ + 2

= 3(1 - (√3/3)²) + 2 cos θ/(3 cos θ) + 2 = 3(1 - 3/9) + 2 cos θ/(3 cos θ) + 2

= 3(1 - 1/3) + 2 cos θ/(3 cos θ) + 2

= 3(2/3) + 2 cos θ/(3 cos θ) + 2

= 2 + 2 cos θ/(3 cos θ) + 2

= 4 + 2 cos θ/(3 cos θ)

Now, since cos θ/(3 cos θ) is equal to 1/3 (cos θ cancels out), we can simplify the expression further:

4 + 2 cos θ/(3 cos θ)

= 4 + 2(1/3)

= 4 + 2/3

= 12/3 + 2/3

= 14/3