Question

If the sum of all solutions of equation sinx + 2cosx = `1 + sqrt(3)cosx` in [0, 2π] is `(kπ)/6`. The value of k is ______.

Options

• 4.00

• 5.00

• 6.00

• 7.00

Answer 1

If the sum of all solutions of equation sinx + 2cosx = `1 + sqrt(3)cosx` in [0, 2π] is `(kπ)/6`. The value of k is 5.00.

Explanation:

`sinx + (2 - sqrt(3))cosx` = 1

`sinx + (sin  π/12)/(cos  π/12) cosx` = 1

⇒ `cos  π/12sinx + sin  π/12cosx = cos  π/12`

⇒ `sin(x + π/12) = sin((5π)/12)`

⇒ `x + π/12 = (5π)/12` or `x + π/12 = π - (5π)/12`

⇒ x = `π/3` or `π/2`

⇒ Sum of all values i.e. `(kπ)/6 = (5π)/6`

⇒ k = 5