# Arcs/Arc Length

• Nov 16th 2009, 11:30 PM
goliath
Arcs/Arc Length
Hey guys, another question for ya:

0 = theda

Suppose that 0 = 150 degrees

a. Find the length of the circular arc corresponding to 0(theda) on a circle of radius 5.
(For this question I got, 13.09, so could someone check to see if it's correct?)

b. Find the area of the slice corresponding to 0(theda) on a circle of radius 5

c. Find sin 0 and cos 0 (again 0 = theda)

d. Find the angle that the radial line segment of the unit circle ending at (sin 0, cos 0) makes with the positive horizontal axis.

I would GREATLY appreciate it if you guys could also provide the answers just for proofing and rechecking.
• Nov 17th 2009, 05:19 AM
Hello goliath
Quote:

Originally Posted by goliath
Hey guys, another question for ya:

0 = theda

Suppose that 0 = 150 degrees

a. Find the length of the circular arc corresponding to 0(theda) on a circle of radius 5.
(For this question I got, 13.09, so could someone check to see if it's correct?)

b. Find the area of the slice corresponding to 0(theda) on a circle of radius 5

c. Find sin 0 and cos 0 (again 0 = theda)

d. Find the angle that the radial line segment of the unit circle ending at (sin 0, cos 0) makes with the positive horizontal axis.

I would GREATLY appreciate it if you guys could also provide the answers just for proofing and rechecking.

If $\displaystyle \theta$ is measured in radians, then the formulae for arc length, $\displaystyle s$, and area of sector, $\displaystyle A$, are:
$\displaystyle s =r\theta$

$\displaystyle A = \tfrac12r^2\theta$
In this question $\displaystyle \theta = 150^o=\frac{5\pi}{6}$ radians and $\displaystyle r = 5$.

So:

(a) Arc length $\displaystyle s = r\theta=5\times \frac{5\pi}{6}\approx 13.09\text{ cm}$. So you are right!

(b) Area of sector $\displaystyle A = \tfrac12r^2\theta=\frac{5^2\cdot5\pi}{2\cdot6}\app rox32.7\text{ cm}^2$

(c) $\displaystyle \sin\left(\frac{5\pi}{6}\right)=\sin\left(\pi - \frac{5\pi}{6}\right)=\sin\left(\frac{\pi}{6}\righ t)=\frac12$

$\displaystyle \cos\left(\frac{5\pi}{6}\right)=-\cos\left(\pi - \frac{5\pi}{6}\right)=-\cos\left(\frac{\pi}{6}\right)=-\frac{\sqrt3}{2}$

(d) We want the point whose coordinates are $\displaystyle \left(\frac12,-\frac{\sqrt3}{2}\right)$. Sketch a diagram, then, and you'll see that the point is in the fourth quadrant, with the radial line making an angle $\displaystyle 60^o$ below the $\displaystyle x$-axis. So the angle we want is $\displaystyle -60^o$ (or $\displaystyle 300^o$), measuring angles positive anticlockwise.