Respuesta :
There are three possible solution (0, π/2, 3π/2) for the given trigonometric equation. So, correct answer is option d.
What is Trigonometric equation?
The trigonometric equations are similar to algebraic equations and can be linear equations, quadratic equations, or polynomial equations. In trigonometric equations, the trigonometric ratios of Sinθ, Cosθ, Tanθ are represented in place of the variables, as in a normal polynomial equation.
So, first of all, we need to figure the angles that cannot be included in our answers out. The only function in the equation that isn't defined for some angles is tan(θ/2) so let's focus on that part of the equation first.
We know that,
tan (θ/2) = {Sin (θ/2)} / {cos (θ/2)}
therefore, cos (θ/2) ≠ 0
so we need to find the angles that will make the cos function equal to zero. So we get:
cos (θ/2) = 0
θ/2 = cos⁻¹0
θ/2 = π/2 + nπ
θ = π + 2nπ
we can now start plugging values in for n:
θ = π + 2π X 0
θ = π
if we plugged any value greater than 0, we would end up with an angle that is greater than 2π so, that's the only angle we cannot include in our answer set, so:
θ ≠ π
having said this, we can now start solving the equation:
tan θ/2 = sin θ
we can start solving this equation by using the half angle formula, such a formula tells us the following:
tan θ/2 = (1 - cos θ)/sin θ
so we can substitute it into our equation:
(1 - cos θ)/sin θ = sin θ
1 - cos θ = sin²θ
1 - cos θ = 1 - cos²θ
cos θ = cos²θ
cos²θ - cos θ = 0
cos θ ( cos θ - 1) = 0
cos θ = 0 or cos θ -1 = 0
θ = π/2, 3π/2 θ = 0
Thus, There are three possible solution (0, π/2, 3π/2) for the given trigonometric equation. So, correct answer is option d.
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