cos2x - 5sinx - 3 = 0
Answers
Step-by-step explanation:
cos(2x) - 5sin(x) - 3 = 0
(cos(x))^2 - (sin(x))^2 - 5sin(x) - 3 = 0
(1 - (sin(x))^2) - (sin(x))^2 - 5sin(x) - 3 = 0
-2(sin(x))^2 - 5sin(x)) -2 = 0
2(sin(x))^2 + 5sin(x) + 2 = 0
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(2sin(x)) + 1)(sin(x) + 2) = 0
solution occurs when 2sin(x) + 1 = 0... sin(x) + 2 can never equal 0 since that would require sin(x) = -2 (that is not possible)...
2sin(x) + 1 = 0
2sin(x) = - 1
sin(x) = -1/2
between 0 and 360 degrees, this shall be satisfied at 210 degrees and 330 degrees... more generally, the solution could be expressed in degrees as (210 + 360(n-1)) degrees and (330 + 360(n-1)) degrees where n runs from 1,2,3.........
cos(2x) - 5sin (x) - 3 = 0
⇒ 1 - 2 sin² (x) - 5 sin (x) - 3 = 0
⇒ - 2 sin² (x) - 5 sin (x) - 2 = 0
⇒ - 2 sin² (x) - 5 sin (x) - 2 = 0
⇒ 2 sin² (x) + 5 sin (x) + 2 = 0
⇒ 2 sin² (x) + sin (x) + 4 sin (x) + 2 = 0
⇒ sin (x) [2 sin (x) + 1] + 2 [2 sin (x) + 1] = 0
⇒ [2 sin (x) + 1] [sin (x) + 2] = 0
⇒ sin (x) = -1/2 ; sin (x) = -2
But since sin (x) ∈ [-1, 1], sin (x) ≠ 2, because -2 ∉ [-1, 1].
So, sin (x) = -1/2.
Since sin (x) is negative, x lies in third or fourth quadrant.
We know sin (π/6) = 1/2. Consider this one.
We will get RHS as -1/2 if we add π or 3π/2 to π/6. → (1)
We will get RHS as -1/2 if we subtract π/2 or π from π/6 too. → (2)
From (1) and (2), we can see a common thing.
sin (π/6 ± π) = -1/2
On adding or subtracting 2π continuously to/from π/6 ± π, the RHS won't change.
sin (π/6 ± π ± 2π ± 2π ± 2π ±...) = -1/2
We simply write this as,
sin (π/6 + (2n - 1)π) = -1/2 ∀n ∈ Z.
From this we get,
x = π/6 + (2n - 1)π = (2n - 5/6)π, ∀n ∈ Z.
From (1) and (2), we can see another common thing.
sin (π/6 + π ± π/2) = -1/2
⇒ sin (7π/6 ± π/2) = -1/2
As we did earlier, on adding or subtracting 2π continuously to/from 7π/6 ± π/2, the RHS won't change.
sin (7π/6 ± π/2 ± 2π ± 2π ± 2π ±...) = -1/2
We simply write this as,
sin (7π/6 + (4n + 1)π/2) = -1/2, ∀n ∈ Z.
From this, we get,
x = 7π/6 + (4n + 1)π/2 = (6n + 5)π/3, ∀n ∈ Z.
So, two possible values for x are found.
1. x = (2n - 5/6)π
2. x = (6n + 5)π/3