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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.

Read in the blogs more about the unique learning experience at nubtrek.continue

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*summary of this topic*

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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

*plain and simple summary*

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*plain and simple summary*

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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.

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

• `cos (270-theta) = - sin theta`

• `tan (270-theta) = cot theta`

*comprehensive information for quick review*

Jogger

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Jogger

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*practice questions to master the knowledge*

Exercise

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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