Thread: How to calculate arc length in unit circle

1. How to calculate arc length in unit circle

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

2. Re: How to calculate arc length in unit circle

Can we not just use the fact that $\displaystyle \tan\theta=\frac{y}{x}$?

3. 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:
$\displaystyle \tan(\theta)=\frac{y}{x}$

4. Re: How to calculate arc length in unit circle

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 $\displaystyle x~\&~y$.

5. Re: How to calculate arc length in unit circle

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

6. Re: How to calculate arc length in unit circle

Originally Posted by Prove It
As long as the angle is measured in radians
What else are? Angular measures are numbers (radians)?

7. Re: How to calculate arc length in unit circle

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...