What is the general solution for cos(4x)+sin(x/2)=0 ?

The general solution for cos(4x)+sin(x/2)=0 is No Solution for x\in\mathbb{R}

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cos(4x)+sin(x/2)=0

cos (4 x) + sin (\frac{x}{2}) = 0

No Solution for x \in R

Solution steps

cos (4 x) + sin (\frac{x}{2}) = 0

Subtract

sin (\frac{x}{2})

from both sides

cos (4 x) = - sin (\frac{x}{2})

Rewrite using trig identities

sin (\frac{π}{2} - 4 x) = sin (- (\frac{x}{2}))

Apply trig inverse properties

- (\frac{x}{2}) = \frac{π}{2} - 4 x + 2 πn, - (\frac{x}{2}) = π - (\frac{π}{2} - 4 x) + 2 πn

- (\frac{x}{2}) = \frac{π}{2} - 4 x + 2 πn : x = \frac{π + 4 πn}{7}

- (\frac{x}{2}) = π - (\frac{π}{2} - 4 x) + 2 πn : x = - \frac{π + 4 πn}{9}

Since the equation is undefined for:

\frac{π + 4 πn}{7}, - \frac{π + 4 πn}{9}

No Solution for x \in R

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What is the general solution for cos(4x)+sin(x/2)=0 ?
The general solution for cos(4x)+sin(x/2)=0 is No Solution for x\in\mathbb{R}