Understanding Tangent and Cotangent

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Discussion Overview

The discussion revolves around the understanding of the tangent and cotangent functions in trigonometry, including their definitions, relationships, and specific examples involving coordinates on the unit circle. The scope includes mathematical reasoning and conceptual clarification.

Discussion Character

  • Mathematical reasoning
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants propose that tangent is defined as y/x, using the example of √3/3 corresponding to the coordinates (√3/2, 1/2).
  • Others assert that cotangent is also defined as x/y and question the conditions under which cotangent equals zero, suggesting that this occurs when cos(θ) equals zero.
  • One participant raises the point that when taking a root, both positive and negative values should be considered on the unit circle.
  • Another participant challenges the relevance of considering both signs in the context of the specific example provided.

Areas of Agreement / Disagreement

Participants express differing views on the relevance of considering both positive and negative roots in the context of the tangent and cotangent functions, indicating that the discussion remains unresolved.

Contextual Notes

There are unresolved mathematical steps regarding the calculations of y/x and the implications of cotangent equaling zero, which depend on the definitions and conditions discussed.

Tyrion101
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I believe I understand both, for instance tangent is equal to y/x and so when you have something like root3/3 it would be equal to the points (root3/2 (x coordinate), and 1/2(y coordinate.) For Cotangent equal to zero, would be (0,1) and (0, -1) am I off base here? I think also that Cot is equal to x/y.
 
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Tyrion101 said:
I believe I understand both, for instance tangent is equal to y/x and so when you have something like root3/3 it would be equal to the points (root3/2 (x coordinate), and 1/2(y coordinate.)

You can check your calculations here.

tan θ = y / x

If x = (√3) / 2 and y = 1 / 2, then what is y / x = ?

For Cotangent equal to zero, would be (0,1) and (0, -1) am I off base here? I think also that Cot is equal to x/y.

By definition, cot (θ) = 1 / tan (θ) = cos (θ) / sin (θ)

In order for cot (θ) = 0, this implies that cos (θ) = 0

For which values of θ does cos (θ) = 0? (Hint: check using the unit circle.)
 
When you take a root, you want both positive and negative answers on the unit circle yes?
 
Tyrion101 said:
When you take a root, you want both positive and negative answers on the unit circle yes?
That's irrelevant in this case.

Your specific example involved x = (√3) / 2 and y = 1 / 2.

If you do the arithmetic to calculate y / x, what number do you get?
 

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