Wondering exactly what sine is.

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Discussion Overview

The discussion revolves around the mathematical concept of sine, its definition, and its relationship to angles, particularly in the context of the unit circle and historical calculations of sine values without calculators. Participants explore various interpretations and methods related to sine and its applications in trigonometry.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether sine can be viewed as a constant that relates to the opposite/hypotenuse ratio, expressing curiosity about historical methods of calculating sine values without calculators.
  • Another participant suggests that sine can be interpreted as the y-coordinate of a point on the unit circle corresponding to a given angle, with cosine as the x-coordinate and tangent as the ratio of y to x.
  • A participant describes methods for calculating sine values for standard angles using algebra and interpolation, mentioning specific angles like 30, 45, and 72 degrees, and referencing the addition and subtraction formulas for angles.
  • One participant provides a table of sine and cosine values for standard angles, indicating the sine values for angles 0, 30, 45, 60, and 90 degrees.
  • There is a correction regarding the definition of tangent at 90 degrees, with one participant asserting that it is undefined, while another acknowledges the correction.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of sine and its calculation methods. There is no consensus on a singular definition or method, and some points remain contested, particularly regarding the nature of tangent at 90 degrees.

Contextual Notes

Some discussions involve assumptions about the definitions of trigonometric functions and the methods used for calculating sine values historically, which may not be fully explored or agreed upon.

repugno
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I was wondering exactly what sine is. i know that the relationship between sine and the angle is expressed as sinx=opposite/hypotoneuse. But is sine somekind of constant you multiply the angle with to get the O/H ratio? How did the people calculate the ratios without calculaters when they only knew the angles?
 
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Can't sine theta be thought of as the y value of a point on the unit circle when you rotate it through theta?
cosine is the x value, tan is y/x.

Hence, sine 90 is 1, cos 90 is zero. tan 90 is indeterminant (1/0)

-Gareth
 
How did the people calculate the ratios without calculaters when they only knew the angles?

Use algebra to find a lot of values, then interpolate to fill in the gaps.

You should be able to do sin 30 and sin 45 without a calculator. :-p sin 72 can be done with pentagons. (same with cosine) Every multiple of 3 can then be gotten through the addition and subtraction of angles formulae. The half angle formuale can get you even finer... (and the one/third angle formulae, if you know the cubic formula)

One way your calculator could compute them is through their MacLauren series... but I understand they actually use a much cleverer method.


tan 90 is indeterminant (1/0)

No, tan 90 is undefined -- 90 is not in the domain of the tangent function.
 
you can learn the values of some standard angles
Code: 
degree 0    30       45                      60                   90
 Sin     0    1/2      1/squareroot 2     (root 3)/ 2         1
Cos    reverse the series
 
Last edited:
Hurkyl said:
No, tan 90 is undefined -- 90 is not in the domain of the tangent function.
Oops! Thanks Hurkyl :redface:
 

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