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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 the 2nd quadrant equivalently as trigonometric ratios of acute angle in the 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 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`
  • `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`
  • `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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