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Thought-Process to Discover Knowledge

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nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

nub,

trek,

jogger,

exercise.

User Guide

nub is the simple explanation of the concept.

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User Guide

trek is the step by step exploration of the concept.

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User Guide

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User Guide

exercise provides practice problems to become fluent in the concepts.

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge.

summary of this topic

### Trigonometric Ratio for Angles in All Quadrants

Voice

Voice

Home

»  270+theta
→  theta measured with y axis
→  complementary functions

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

plain and simple summary

nub

plain and simple summary

nub

dummy

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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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.

Keep tapping on the content to continue learning.
Starting on learning representing trigonometric ratios for "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.

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

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
• 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

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Progress

Progress

The angle omega equals 270 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 y and p y = x. What is tan 270 plus 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 'minus x by y equals minus cot theta'
For angles given as 270 plus theta, the trigonometric ratios are: sine 270 plus theta equals minus cos theta;; cos 270 plus theta equals sine theta;; tan 270 plus theta equals minus cot theta
The angle omega equals 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 = x and p y = minus y. What is tan minus theta?
equals tan
y by x equals tan theta
minus tan
minus y by x equals minus tan theta
equals cot
x by y equals cot theta
minus cot
x by minus y equals minus cot theta
The answer is ' minus y by x equals minus tan theta'
For angles given as minus theta, the trigonometric ratios are: sine minus theta equals minus sine theta;; cos minus theta equals cos theta;; tan minus theta equals minus tan theta

we are not perfect yet...