__maths__>__Advanced Trigonometry__>__Trigonometric 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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