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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.

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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.

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. 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.

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. 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.

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. sin omega = (-b)/1

cos omega = a/1

tan omega = (-b)/a

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. 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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