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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.

Read in the blogs more about the unique learning experience at nubtrek.continue
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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trigonometric ratio of angle > 90 The angle `omega` is shown in the figure. The magnitude of projections on x axis and y axis is given as a and b.

It is noted that the projection along x axis is `-a` and so

`tan theta = 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)`

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

Considering the projections along x axis and y axis as `-a` and `-b`,

`tan omega = (-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 > 180 `sin omega = (-b)/1`

`cos omega = (-a)/1`

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

trigonometric ratio of negative angle 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.

Considering the projections on x axis and y axis as `a` and `-b`, `tan omega = (-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`

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

s `tan 590^@ = (-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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