What is the general solution for sin(x/2)=-cos(x-pi/3) ?

The general solution for sin(x/2)=-cos(x-pi/3) is No Solution for x\in\mathbb{R}

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sin(x/2)=-cos(x-pi/3)

sin (\frac{x}{2}) = - cos (x - \frac{π}{3})

No Solution for x \in R

Solution steps

sin (\frac{x}{2}) = - cos (x - \frac{π}{3})

Multiply by

- 1

- sin (\frac{x}{2}) = cos (x - \frac{π}{3})

Rewrite using trig identities

sin (- (\frac{x}{2})) = sin (\frac{π}{2} - (x - \frac{π}{3}))

Apply trig inverse properties

- (\frac{x}{2}) = \frac{π}{2} - (x - \frac{π}{3}) + 2 πn, - (\frac{x}{2}) = π - (\frac{π}{2} - (x - \frac{π}{3})) + 2 πn

- (\frac{x}{2}) = \frac{π}{2} - (x - \frac{π}{3}) + 2 πn : x = \frac{5 π + 12 πn}{3}

- (\frac{x}{2}) = π - (\frac{π}{2} - (x - \frac{π}{3})) + 2 πn : x = - \frac{π + 12 πn}{9}

Since the equation is undefined for:

\frac{5 π + 12 πn}{3}, - \frac{π + 12 πn}{9}

No Solution for x \in R

3 = cot^{2} (θ) + 4 cot (θ)

cos^{2} (2 x) + 3 sin (2 x) - 3 = 0

1 - cos^{2} (x) = 0.75 1 - cos^{2} (y)

sec^{2} (x) - 6 tan (x) + 4 = 0

1 - cos (x) = 2 - 2 sin^{2} (x)

What is the general solution for sin(x/2)=-cos(x-pi/3) ?
The general solution for sin(x/2)=-cos(x-pi/3) is No Solution for x\in\mathbb{R}