Solve 4(cos(pi/3)+sin(pi/3))=xsin(θ)

$$4( \cos ( \frac{ \pi }{ 3 } ) + \sin ( \frac{ \pi }{ 3 } ) )=x \sin ( \theta )$$

Solve for x

$$x=\frac{2\left(\sqrt{3}+1\right)}{\sin(\theta )}$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}$$

Steps for Solving Linear Equation

Get the value of \cos(\frac{\pi }{3}) from trigonometric values table.

$$4\left(\frac{1}{2}+\sin(\frac{\pi }{3})\right)=x\sin(\theta )$$

Get the value of \sin(\frac{\pi }{3}) from trigonometric values table.

$$4\left(\frac{1}{2}+\frac{\sqrt{3}}{2}\right)=x\sin(\theta )$$

Since \frac{1}{2} and \frac{\sqrt{3}}{2} have the same denominator, add them by adding their numerators.

$$4\times \frac{1+\sqrt{3}}{2}=x\sin(\theta )$$

Cancel out 2, the greatest common factor in 4 and 2.

$$2\left(1+\sqrt{3}\right)=x\sin(\theta )$$

Use the distributive property to multiply 2 by 1+\sqrt{3}.

$$2+2\sqrt{3}=x\sin(\theta )$$

Swap sides so that all variable terms are on the left hand side.

$$x\sin(\theta )=2+2\sqrt{3}$$

The equation is in standard form.

$$\sin(\theta )x=2\sqrt{3}+2$$

Divide both sides by \sin(\theta ).

$$\frac{\sin(\theta )x}{\sin(\theta )}=\frac{2\sqrt{3}+2}{\sin(\theta )}$$

Dividing by \sin(\theta ) undoes the multiplication by \sin(\theta ).

$$x=\frac{2\sqrt{3}+2}{\sin(\theta )}$$

Divide 2+2\sqrt{3} by \sin(\theta ).

$$x=\frac{2\left(\sqrt{3}+1\right)}{\sin(\theta )}$$

Solve for θ

$$\theta =arcSin(2\left(1+3^{\frac{1}{2}}\right)x^{-1})+2\pi n_{1}\text{, }n_{1}\in \mathrm{Z}$$

$$\theta =\pi +2\pi n_{2}+\left(-1\right)arcSin(2x^{-1}\left(1+3^{\frac{1}{2}}\right))\text{, }n_{2}\in \mathrm{Z}$$