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Trigonometric Ratio for Angles in All Quadrants

Trigonometric Ratio for Angles in All Quadrants

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From 3rd to 1st Quadrant


 »  `180+theta`
    →  `theta` measured with x axis
    →  equivalent functions


 »  `270-theta`
    →  `theta` measured with y axis
    →  complementary functions

Angles in 3rd Quadrant

plain and simple summary

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plain and simple summary

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simple steps to build the foundation

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The results of representing trigonometric ratios of angles in 3rd quadrant equivalently as trigonometric ratios of acute angle in 1st quadrant are explained.


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Starting on learning representing trigonometric ratios for "Angles in 3rd Quadrant". ;; The results of representing trigonometric ratios of angles in 3rd quadrant equivalently as trigonometric ratios of acute angle in 1st quadrant are explained.

The angle `omega = 180+theta` is shown in the figure as point P. The similar triangle with angle theta is given by `Q(x,y)` in 1st quadrant. trigonometric ratio of angle 180+theta Note: Given `Q(x,y)`, the coordinates of `P` is `(color(coral)(P_x),color(deepskyblue)(P_y))`:
`color(coral)(P_x=-x)` and `color(deepskyblue)(P_y=-y)`

What is `tan (180+theta)`?

  • `(-y)/(-x) = tan theta`
  • `y/(-x) = - tan theta`
  • `(-x)/(-y)= cot theta`
  • `(-x)/y = - cot theta`

The answer is '`-y/-x = tan theta`'

`tan (180+theta)` (tan of the given angle)
`quad = (P_y)/(P_x)` (by definition of `tan`)
`quad = (-y)/(-x)` (substituting the values)
`quad = tan theta` (equivalently in 1st quadrant.)

Note: learners can work out this for `sin` and `cos`.

For angles given as `180+theta`, the trigonometric ratios are:trigonometric ratio of angle 180+theta  • `sin (180+theta) = - sin theta`

 • `cos (180+theta) = - cos theta`

 • `tan (180+theta) = tan theta`

    The angle `omega = 270-theta` is shown in the figure as point P. The similar triangle with angle theta is given by `Q(x,y)` in 1st quadrant. trigonometric ratio of angle 270-theta Note: Given `Q(x,y)`, the coordinates of `P` is `(color(coral)(P_x),color(deepskyblue)(P_y))`:
    `color(coral)(P_x=-y)` and `color(deepskyblue)(P_y=-x)`

    What is `tan (270-theta)`?

    • `(-y)/(-x) = tan theta`
    • `y/(-x) = - tan theta`
    • `(-x)/(-y)= cot theta`
    • `(-x)/y = - cot theta`

    The answer is '`(-x)/(-y)= cot theta`'

    `tan (270-theta)` (tan of the given angle)
    `quad = (P_y)/(P_x)` (by definition of `tan`)
    `quad = (-x)/(-y)` (substituting the values)
    `quad = cot theta` (equivalently in 1st quadrant.)

    Note: learners can work out this for `sin` and `cos`.

    For angles given as `270-theta`, the trigonometric ratios are:trigonometric ratio of angle 270-theta  • `sin (270-theta) = - cos theta`

     • `cos (270-theta) = - sin theta`

     • `tan (270-theta) = cot theta`

      comprehensive information for quick review

      Jogger

      comprehensive information for quick review

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      practice questions to master the knowledge

      Exercise

      practice questions to master the knowledge

      Exercise

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      The angle omega equals 180 plus theta is shown in the figure as point P. The similar triangle with angle theta is given by Q x y in 1st quadrant ;; Note: Given Q x y , the coordinates of P is P x = minus x and p y = minus y. What is tan 180 plus theta?
      equals tan
      minus y by minus x equals tan theta
      minus tan
      y by minus x equals minus tan theta
      equals cot
      minus x by minus y equals cot theta
      minus cot
      minus x by y equals minus cot theta
      The answer is ' minus y by minus x equals tan theta'
      For angles given as 180 plus theta, the trigonometric ratios are: sine 180 plus theta equals minus sine theta;; cos 180 plus theta equals minus cos theta;; tan 180 plus theta equals tan theta
      The angle omega equals 270 minus theta is shown in the figure as point P. The similar triangle with angle theta is given by Q x y in 1st quadrant ;; Note: Given Q x y , the coordinates of P is P x = minus y and p y = minus x. What is tan 270 minus theta?
      equals tan
      minus y by minus x equals tan theta
      minus tan
      y by minus x equals minus tan theta
      equals cot
      minus x by minus y equals cot theta
      minus cot
      minus x by y equals minus cot theta
      The answer is ' minus x by minus y equals cot theta'
      For angles given as 270 minus theta, the trigonometric ratios are: sine 270 minus theta equals minus cos theta;; cos 270 minus theta equals minus sine theta;; tan 270 minus theta equals cot theta

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