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Trigonometric Values for all angles in unit circle form

Trigonometric Values for all angles in unit circle form

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Trigonometric Values : First Principles


 »  The angle `omega` is measured from positive x-axis
    →  x and y coordinates takes sign

Quickly follow the sign of x and y projections to find sign of trigonometric values. No need to memorize.

 »  `P` is in the second quadrant
    →  -ve x projection
    →  +ve y projection


 »  `Q` is in third quadrant
    →  -ve: both x and y projections


 »  `R` is in fourth quadrant
    →  +ve x projection
    →  -ve y projection

Trigonometric values for Any Angle: First Principles

plain and simple summary

nub

plain and simple summary

nub

dummy

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.

simple steps to build the foundation

trek

simple steps to build the foundation

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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.


Keep tapping on the content to continue learning.
Starting on learning "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.

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)`

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)`

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`

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)`

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)`

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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))`



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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Progress

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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. ;; What is tan omega?
by a
b by ae
minus;-
b by minus a
The answer is 'b by minus 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 square root a square plus b square equals 1. ;; sine omega ; equals ; b by 1 ;; divided by one is given to specify the radius of the circle. ;; cos omega ; equals ; minus a by 1 ;; tan omega ; equals ; b by minus 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. ;; What is tan omega?
by a
b by a
minus;-
minus b by minus a
The answer is 'minus b by minus 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.;; sine omega ; equals ; minus b by 1 ;; divided by one is given to specify the radius of the circle. ;; cos omega ; equals ; minus a by 1 ;; tan omega ; equals ; minus b by minus 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. ;; What is tan omega?
by a
b by a
minus;-
minus b by a
The answer is ' minus b by a'
The trigonometric ratios are computed from the x and y axis projections. ;; sine omega ; equals ; minus b by 1 ;; cos omega ; equals ; a by 1 ;; tan omega ; equals ; minus b by 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. ;; What is tan omega?
by b;a by
a by b
minus;-
minus b by minus a
The answer is 'minus b by minus a'
The trigonometric ratios are computed from the x and y axis projections.;; sine omega ; equals ; minus b by 1 ;; cos omega ; equals ; minus a by 1 ;; tan omega ; equals ; minus b by minus 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 ;; sine omega is y projection ;; cos omega is x projection;; tan omega is y projection by x projection.

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