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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.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

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The results of representing trigonometric ratios of angles in 4th quadrant equivalently as trigonometric ratios of acute angle in 1st quadrant are explained.

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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 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 (90-theta)`?

  • `y/x = tan theta`
  • `y/(-x) = - tan theta`
  • `x/y= cot 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 = y/x` (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 trigonometric ratios are:trigonometric ratio of angle 90-theta  • `sin (90-theta) = cos theta`

 • `cos (90-theta) = sin theta`

 • `tan (90-theta) = cot theta`

For angles measured in reference to x axis, the trigonometric ratios remain the same when equivalently represented in the 1st quadrant.

For angles measured in reference to y axis, the trigonometric ratios are equivalently complementary ratios in the 1st quadrant.

Trigonometric Ratios in 1st Quadrant:
For Angles in reference to x-axis:
`sin(180+-theta) = -+sin theta`
`cos(180+-theta) = -cos theta`
`tan(180+-theta) = +-tan theta`

`sin(-theta) = -sin theta`
`cos(-theta) = cos theta`
`tan(-theta) = -tan theta`

For Angles in reference to y-axis:
`sin(90+-theta) = cos theta`
`cos(90+-theta) = -+sin theta`
`tan(90+-theta) = -+cot theta`

`sin(270+-theta) = -cos theta`
`cos(270+-theta) = sin theta`
`tan(270+-theta) = -tan theta`

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