Server Error

Server Not Reachable.

This may be due to your internet connection or the nubtrek server is offline.

Thought-Process to Discover Knowledge

Welcome to nubtrek.

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.

User Guide

Welcome to nubtrek.

The content is presented in small-focused learning units to enable you to
think,
figure-out, &
learn.

Just keep tapping (or clicking) on the content to continue in the trail and learn.

User Guide

To make best use of nubtrek, understand what is available.

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.

This captures the small-core of concept in simple-plain English. The objective is to make the learner to think about.

User Guide

trek is the step by step exploration of the concept.

Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step.

User Guide

jogger provides the complete mathematical definition of the concepts.

This captures the essence of learning and helps one to review at a later point. The reference is available in pdf document too. This is designed to be viewed in a smart-phone screen.

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

»  90-theta
→  theta measured with y axis
→  complementary functions

plain and simple summary

nub

plain and simple summary

nub

dummy

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.

simple steps to build the foundation

trek

simple steps to build the foundation

trek

Support Nubtrek

You are learning the free content, however do shake hands with a coffee to show appreciation.
To stop this message from appearing, please choose an option and make a payment.

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 = 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. 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

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:  • sin (90-theta) = cos theta

• cos (90-theta) = sin theta

• tan (90-theta) = cot theta

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Progress

Progress

The angle omega equals 90 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 = y and p y = x. What is tan 90 minus 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 ' x by y equals cot theta'
For angles given as 90 minus theta, the trigonometric ratios are: sine 90 minus theta equals cos theta;; cos 90 minus theta equals sine theta;; tan 90 minus theta equals 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.
For angles in reference to x axis; the trigonometric ratios sine, cos, tan remain the same as sine, cos, tan when equivalently represented in the first quadrant.;; For angles in reference to y axis; the trigonometric ratios sine, cos, tan are equivalently complementary ratios cos, sine, cot when equivalently represented in the first quadrant.

we are not perfect yet...