__maths__>__Advanced Trigonometry__>__Trigonometric Ratio for Angles in All Quadrants__### Angles in 4th Quadrant

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

*click on the content to continue..*

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`
- `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`

The angle `omega = -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 (-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 (-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 `-theta`, the trigonometric ratios are: • `sin (-theta) = - sin theta`

• `cos (-theta) = cos theta`

• `tan (-theta) = - tan theta`

*switch to slide-show version*