How to calculate arc length in unit circle

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Discussion Overview

The discussion revolves around the method of calculating arc length in a unit circle, specifically focusing on the relationship between known coordinates (x, y) and the arc length. The scope includes mathematical reasoning and conceptual clarification regarding the use of formulas and trigonometric knowledge.

Discussion Character

  • Mathematical reasoning, Conceptual clarification

Main Points Raised

  • One participant asks for the method to calculate arc length given known coordinates (x, y).
  • Another participant notes that calculators and trigonometric tables are not allowed in this context.
  • A participant presents the formula for arc length as s = r θ, emphasizing that θ should be in radians.
  • It is mentioned that the original poster (OP) seems to understand this formula, as they indicate that arc length equals θ when r = 1.
  • There is a discussion about the usefulness of knowing trigonometric values, particularly in relation to special angles and triangles, with an example provided for specific values of x and y.
  • Another participant suggests that if the coordinates do not correspond to special angles, inverse sine may be necessary to find θ.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of knowing trigonometric tables and the conditions under which the formula applies. The discussion remains unresolved regarding the best approach when the coordinates do not align with special angles.

Contextual Notes

There are limitations regarding the assumptions about the coordinates and their relationship to special angles, as well as the dependence on whether the use of inverse functions is acceptable in the calculation.

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http://www.up98.org/upload/server1/01/z/cllb59cvnwaigmmar6b5.jpeg

What is the method of calculating arc length in In the image above .
x & y is known
Thanks .
 
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Feuilleton :
Obviously, the use of calculators and trigonometric tables is not allowed
 
There is a formula for it.

s = r θ, where θ is in radians.

You shouldn't need a Trig table, you should know the table. The angles in the unit circle are special angles which you should know by heart.
 
Ivan92 said:
There is a formula for it.

s = r θ, where θ is in radians.
It looks like the OP already knows this, because outside of the circle he/she writes: "arc length = θ = ?" (since r = 1).

As for knowing the table, that is helpful ONLY if x, y, and r form one of the two special triangles. If x = √3/2 and y = 1/2, then sure, we can find the arc length no trouble. But what if x = 0.6 and y = 0.8? We would need to make use of the inverse sine, wouldn't we?
 

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