Question

How do you solve $2\cos x\sin x + \sin x = 0$

Answer 1

The given is a full circle so the angle is restricted from ${0^0}$to ${360^0}$. The sine wave represents the value of sin with respect to angle. The cosine wave represents the value of cos with respect to angle.
We need to find values for the sine and cos value from the equation. For that, the basic values of sine and cosine must be known.

Complete step by step answer:
Given,
The equation that is needed to solve is $2\cos x\sin x + \sin x = 0$
The equation to be solved is
$2\cos x\sin x + \sin x = 0$.
We can take common out of the equation is
$\sin x(2\cos x + 1) = 0$
We can equal the equation,
$\sin x = 0$
$2\cos x + 1 = 0$
We can take the values of equations, common values are taken as solutions.
First, we must calculate the value for $\sin x = 0$,
The sine wave represents the value of sin with respect to angle. By the sine wave, we can conclude that sin has a value of $0$if the angles are ${0^0},{120^0},{180^0}$and ${240^0}$. If the angles are written in $0,\dfrac{{2\pi }}{3},2\pi$and $\dfrac{{4\pi }}{3}$. The given is a full circle so the angle is restricted from ${0^0}$to ${360^0}$. So the angle is restricted to ${360^0}$.
We can take the value in the left side of the equation to the right side of the equation,
$x = {\sin ^{ - 1}}(0)$
The value $x$depends on the sine value when it is equal to ${0^0}$, sin has a value of $0$if the angles are ${0^0},{120^0},{180^0}$and ${240^0}$. If the angles are written in $0,\pi ,2\pi$
$x = 0$or or $x = \pi$or $x = 2\pi$
Secondly the value of $2\cos x + 1 = 0$
The terms in the left side of the equation to the right side of the equation,
$2\cos x = - 1$
Divide the values in the left side of the equation to the right side of the equation,
$\cos x = - \dfrac{1}{2}$
We can take the value in the left side of the equation to the right side of the equation,
$x = {\cos ^{ - 1}}\left( { - \dfrac{1}{2}} \right)$
The value of cosine angles $0,30,45,60$and $90$are $0,\dfrac{{\sqrt 3 }}{2},\dfrac{1}{{\sqrt 2 }},\dfrac{1}{2}$and $1$.
By substituting the correct value,
$x = 120$or $x = 360$
Converting the degree into radians, we need to multiply by $\dfrac{\pi }{{180}}$
$x = 120 \times \dfrac{\pi }{{180}}$ or $x = 360 \times \dfrac{\pi }{{180}}$
Multiplying the above terms,
$x = \dfrac{{2\pi }}{3}$or $x = 2\pi$
The value of x is $x = 0,\pi ,2\pi ,\dfrac{{2\pi }}{3}$.

Note: The value must be noted carefully. Always remember that the full cycle given the angle must be taken between ${0^0}$to ${360^0}$It should not be beyond that. Always take the correct value for the sine and cosine. The value must be taken correctly so that the union will not be a mistake. Don’t just leave the answer, always remember to write the value which is both terms.