# Graph of $y=4\sin\left(\frac{x}{3}\right)\cos\left(\frac{x}{3}\right)$

What is the graph of the equation $$y=4\sin\left(\frac{x}{3}\right)\cos\left(\frac{x}{3}\right)$$ along with its period and amplitude?

I got confused when the equation got cosine and sine in it. I only know how to handle graphs of an equation when only there is either cosine or sine, not altogether with sine and cosine in one equation.

• Hint:$\sin 2A=2\sin A \cos A$. – Anurag A Jun 12 '17 at 11:40
• $y=4 \sin(x/3) \cos(x/3) = 2 \sin(2x/3)$ So the amplitude is $2$ and the period id $3\pi$ ... now check it ... desmos.com/calculator/kcoysbbfip – Donald Splutterwit Jun 12 '17 at 11:46
• Please read this tutorial on how to typeset mathematics on this site. – N. F. Taussig Jun 12 '17 at 11:50

Using the following double-angle formula makes graphing your curve a lot easier: $$\sin(2\theta)\equiv 2\sin{\theta}\cos{\theta}$$ Hence, the equation of your curve is equivalent to: $$y=4\sin\left(\frac{x}{3}\right)\cos\left(\frac{x}{3}\right)=2\cdot \color{green}{2\sin\left(\frac{x}{3}\right)\cos\left(\frac{x}{3}\right)}=2\color{green}{\sin\left(\frac{2x}{3}\right)}$$ Given a sinusoid $y=A\sin(\omega x+\varphi)$, the amplitude is $A$ and the period is $\dfrac{2\pi}{\omega}$.