Cot y/2 - sin y=0

Question

Answers between 0 and 2pi

Answer 1

I can use the substitution u=y/2 and apply the double angle formula

cot (y/2)-siny = 0

cot u - sin(2u) = 0

cot u = sin (2u)

(cos u/sin u)= 2 sin u cos u

cos u = 2 sin² u cos u

cos u - 2 sin² u cos u = 0

cos u (1 - 2 sin² u) = 0

cos u = 0 or 1 - 2 sin² u = 0

cos u = 0 or 1/2 = sin² u

cos (y/2) = 0 or sin (y/2) =±√(1/2)

u=y/2 = π/2 or 3π/2 or u =π/6, 5π6, 7π/6,11π/6

y = π or π/3 or 5π/3 (other answers are outside the interval)

Answer 2

coty/2 - siny = 0; cosy/2/siny/2 - 2siny/2cosy/2 = 0; cosy/2(1/siny/2 - 2siny/2) = 0; cosy/2 = 0; y/2 = π/2, y = π

1/siny/2 - 2siny/2 = 0; 1 - 2sin²y/2 = 0; 1 -1 + cosy = 0; cosy = 0; y = π/2 or y = 3π/2

Answer: {π/2, π, 3π/2}

Answer 3

To start with, since cot(x) = cos(y/2)/sin(y/2), this equation is not defines when sin(y/2) = 0. That means y = 0 and y = 2π are not possible solutions. Let's just keep that in the back of our heads.

Since cot(x) = cos(x)/sin(x) we can re-write the problem as:

cos(y/2)/sin(y/2) - sin(y) = 0

Let θ = y/2 therefore, we can re-write the problem as cos(θ)/sin(θ) - sin(2θ) = 0

Using the identity sin(2x) = 2sin(x)cos(x) we can say:

cos(θ)/sin(θ) - 2sin(θ)cos(θ) = 0

Getting a common denominator:

cos(θ)/sin(θ) - 2sin²(θ)cos(θ)/sin(θ) = 0

(cos(θ) - 2sin²(θ)cos(θ))/sin(θ) = 0

The only way the expression "(cos(θ) - 2sin²(θ)cos(θ))/sin(θ)" can n=be equal to zero is if the numerator equals zero, therefore, setting the numerator equal to zero:

cos(θ) - 2sin²(θ)cos(θ) = 0

cos(θ)[1 - 2sin²(θ)] = 0

So either cos(θ) = 0 meaning θ = π/2 and 3π/2

OR 1 - 2sin²(θ) = 0 which means sin²(θ) = 1/2 or sin(θ) = ±√1/2 = ±√2/2 therefore θ = π/4, 3π/4, 5π/4, 7π/4

But we are solving for y, and y/2 = θ, therefore:

y/2 = π/2 so y = π

y/2 = 3π/2 so y = 3π (out of the domain of 0 to 2π)

y/2 = π/4 so y = π/2

y/2 = 3π/4 so y = 3π/2

y/2 = 5π/4 so y = 5π/2 (out of the domain of 0 to 2π)

don't need to check the final answer because it will be outside the domain range.

Final answer: y = π/2, π, and 3π/2