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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

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mathsAdvanced TrigonometryTrigonometric Values for all angles in unit circle form

Trigonometric values for Any Angle: First Principles

This page explains the first principles to calculate a trigonometric ratio for a given angle. The rest of the knowledge extends the first principles to arrive at standard formulas or results.



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The angle `omega` is shown in the figure. The magnitude of projections on x axis and y axis is given as a and b. trigonometric ratio of angle > 90 What is `tan omega`?

  • `b/a`
  • `b/(-a)`
  • `b/(-a)`

The answer is '`b/(-a)`'

The trigonometric ratios are computed from the x and y axis projections.
Note that the projections are given for point on unit circle. so `sqrt(a^2 + b^2) = 1`.trigonometric ratio of angle > 90 `sin omega = b/1`

`cos omega = (-a)/1`

`tan omega = b/(-a)`

The angle `omega` is shown in the figure. The magnitude of projections on x axis and y axis is given as a and b. trigonometric ratio of angle > 180 What is `tan omega`?

  • `b/a`
  • `(-b)/(-a)`
  • `(-b)/(-a)`

The answer is '`(-b)/(-a)`'. Note that the sign of the numerator and denominator provide information as to if the angle is in first quadrant or 3rd quadrant.

The trigonometric ratios are computed from the x and y axis projections.
Note that the projections are given for point on unit circle. so `sqrt(a^2 + b^2) = 1`.trigonometric ratio of angle > 180 `sin omega = (-b)/1`

`cos omega = (-a)/1`

`tan omega = (-b)/(-a)`

The angle `omega` is negative and is shown in the figure. The magnitude of projections on x axis and y axis is given as a and b. trigonometric ratio of negative angle What is `tan omega`?

  • `b/a`
  • `(-b)/a`
  • `(-b)/a`

The answer is '`(-b)/a`'

The trigonometric ratios are computed from the x and y axis projections.
Note that the projections are given for point on unit circle. so `sqrt(a^2 + b^2) = 1`.trigonometric ratio of negative angle `sin omega = (-b)/1`

`cos omega = a/1`

`tan omega = (-b)/a`

The angle `omega = 590` is shown in the figure. The magnitude of projections on x axis and y axis is given as a and b. trigonometric ratio of angle > 180 What is `tan omega`?

  • `a/b`
  • `(-b)/(-a)`
  • `(-b)/(-a)`

The answer is '`(-b)/(-a)`'

The trigonometric ratios are computed from the x and y axis projections.trigonometric ratio of angle > 180 `sin omega = (-b)/1`

`cos omega = (-a)/1`

`tan omega = (-b)/(-a)`

For any angle the trigonometric values are computed using the projections of point on unit circle at the given angle on to x and y axis.

First Principles to find Trigonometric Ratios for any Angle: For the given angle, find the projections of point on unit circle at the given angle on to x and y axes. The projections are signed values. The trigonometric ratios are computed as

 •  `sin omega = y text( projection)`

 •  `cos omega = x text( projection)`

 •  `tan omega = (y text( projection))/(x text( projection))`

                            
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