Question

The length of the arc in one period of the cycloid $x=t+sint,y=1-cost$ is? The values of t run from $0$ to $2\pi$.

• A. $8$
• B. $8\pi$
• C. $4\pi$
• D. $7$

Answer 1

The correct option is A $8$

$x=t+sint$ and $x^{\prime }=1-cost$
$y=1-cost$ and $y^{\prime }=sint$
$\sqrt{x^{\prime 2}+y^{\prime 2}}=\sqrt{(1-cost{)}^2+si{n}^{2}t}=\sqrt{2-2cost}$
Now, use the trigonometric identites, $si{n}^2\left(\frac{t}{2}\right)=\frac{1-cost}{2}$
$\sqrt{x^{\prime 2}+y^{\prime 2}}=2sin\frac{t}{2}$
Length of arc=${\int }_0^{2\pi }2sin\frac{t}{2}dt=(-4cos\frac{2\pi }{2})-(-4cos\frac{0}{2})=4+4=8$