# How to calculate arc length in unit circle

• Dec 7th 2011, 05:58 AM
subuntu
How to calculate arc length in unit circle
Attachment 23030

What is the method of calculating arc length in In the image above .
x & y is known
Thanks .

Feuilleton :
Obviously, the use of calculators and trigonometric tables is not allowed
• Dec 7th 2011, 06:05 AM
Quacky
Re: How to calculate arc length in unit circle
Can we not just use the fact that $\displaystyle \tan\theta=\frac{y}{x}$?
• Dec 7th 2011, 06:06 AM
corsica
Re: How to calculate arc length in unit circle
Recognize that the arc length is the same as the angle between line $\displaystyle r$ and line $\displaystyle x$.

Lines $\displaystyle r$, $\displaystyle x$ and $\displaystyle y$ form a right triangle. Thus you can use the formula:
$\displaystyle \tan(\theta)=\frac{y}{x}$
• Dec 7th 2011, 06:46 AM
Plato
Re: How to calculate arc length in unit circle
Quote:

Originally Posted by subuntu
What is the method of calculating arc length in In the image above .x & y is known

Feuilleton : Obviously, the use of calculators and trigonometric tables is not allowed

Does the question ask for the arc-length from $\displaystyle (1,0)$ to $\displaystyle (x,y)$ in the first quadrant.

If so the answer is $\displaystyle \arccos \left( {\frac{x}{{\sqrt {x^2 + y^2 } }}} \right)$.

That is the measure of the central angle times the radius which in this case is $\displaystyle R=1$.

If it could be that $\displaystyle (x,y)\in II$ the same formula works.

Now if $\displaystyle (x,y)\in III\text{ or }IV$ then we need to know if we are still measuring in a counter-clockwise direction or not.

This directive the use of calculators and trigonometric tables is not allowed is impossible to follow except for a very, very few values of $\displaystyle x~\&~y$.
• Dec 7th 2011, 01:55 PM
Prove It
Re: How to calculate arc length in unit circle
Quote:

Originally Posted by corsica
Recognize that the arc length is the same as the angle between line $\displaystyle r$ and line $\displaystyle x$.

As long as the angle is measured in radians :)
• Dec 7th 2011, 02:14 PM
Plato
Re: How to calculate arc length in unit circle
Quote:

Originally Posted by Prove It
As long as the angle is measured in radians :)

What else are? Angular measures are numbers (radians)?
• Dec 7th 2011, 02:16 PM
Prove It
Re: How to calculate arc length in unit circle
Quote:

Originally Posted by Plato
What else are? Angular measures are numbers (radians)?

It's quite possible that a student might not know much about the radian measure, and seeing a statement like that, would get confused when the arclength is nothing at all like the angle measurement in DEGREES. It was just for clarification...