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Trigonometric Ratio for Angles in All Quadrants

Trigonometric Ratio for Angles in All Quadrants

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From 2nd to 1st Quadrant


 »  `90+theta`
    →  `theta` measured with y axis
    →  complementary functions


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

Angles in 2nd Quadrant

plain and simple summary

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The results of representing trigonometric ratios of angles in the 2nd quadrant equivalently as trigonometric ratios of acute angle in the 1st quadrant are explained.


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Starting on learning representing trigonometric ratios for "Angles in 2nd Quadrant". ;; The results of representing trigonometric ratios of angles in the 2nd quadrant equivalently as trigonometric ratios of acute angle in the 1st quadrant are explained.

The angle `omega = 90+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 90+theta Note: Given `Q(x,y)`, the coordinate 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 (90+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 (90+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 `90+theta`,

 •  the angle is in 2nd quadrant, so x projection is negative and y projection is positive.

 •  the `/_theta` is measured with y-axis, and so the similar triangle in 1st quadrant will have `/_theta` with x-axis, thereby swapping the x and y projections.

the trigonometric ratios are:trigonometric ratio of angle 90+theta  • `sin (90+theta) = cos theta`

 • `cos (90+theta) = - sin theta`

 • `tan (90+theta) = - cot theta`

    The angle `omega = 180-theta` is shown in the figure as point P. The similar triangle with `/_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`

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      The angle omega equals 90 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 point q x, y; the coordinates of p is p x = minus y and p y = x. What is tan 90 plus theta?
      tan
      y by x equals tan theta
      minus tan
      y by minus x equals minus tan theta
      cot
      x by y equals cot theta
      minus cot
      minus x by y equals minus cot theta
      The answer is 'minus x by y equals minus cot theta'
      For angles given as 90 plus theta. the angle is in 2nd quadrant, so x projection is negative and y projection is positive. The angle theta is measured with y-axis, and so the similar triangle in 1st quadrant will have angle theta with x-axis, thereby swapping the x and y projections. the trigonometric ratios are: sine 90 plus theta equals cos theta;; cos 90 plus theta equals minus sine theta;; tan 90 plus theta equals minus cot theta
      The angle omega equals 180 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 x and p y = y. What is tan 180 minus theta?
      equals tan
      y by x equals tan theta
      minus tan
      y by minus x equals minus tan theta
      equals cot
      x by y equals cot theta
      minus cot
      minus x by y equals minus cot theta
      The answer is ' y by minus x equals minus tan theta'
      For angles given as 180 minus theta, the trigonometric ratios are: sine 180 minus theta equals sine theta;; cos 180 minus theta equals minus cos theta;; tan 180 minus theta equals minus tan theta

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