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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 Identities for compound angles

Voice

Voice

Home

sin(A-B), cos(A+B), cos(A-B)

»  sin(A+B) Proven result

Quickly derive the identities. No need to memorize.

»  sin(A-B) = sin(A+(-B))

»  cos(A+B) = sin((90-A) -B)

»  cos(A-B)= sin((90-A)+B)

Geometrical Proof

»  sin(A-B) from bar(RT)=bar(PQ1)
»  cos(A-B) from bar(RS)=bar(PQ)

### Compound Angles: cos(A+B), sin(A-B), cos(A-B)

plain and simple summary

nub

plain and simple summary

nub

dummy

sin(A-B)=sinA cosB - cosA sinB
cos(A-B)=cosA cosB + sinA sinB

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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In this page, a simple proof, based on the earlier result of sin(A+B), is explained for expressing cos(A+B), sin(A-B), cos(A-B) in terms of sinA, sinB, cosA , and cosB.

Keep tapping on the content to continue learning.
Starting on learning "cos A +B, sine A minus B, cos A minus B". ;; In this page, a simple proof based on the earlier result of sine A + B, is explained for expressing cos A + B, sine A minus B, cos A minus B, in terms of sine A, sine B, cos A , and cosB.

It was geometrically proven that cos(A+B)=cosA cosB - sinA sinB. For the same result, we can work out a proof using algebra of trigonometric functions.

cos(A+B)
quad quad = sin(90-A-B)
quad quad = sin[(90-A) + (-B)]
quad quad = sin(90-A)cos(-B)+cos(90-A)sin(-B)
quad quad = cosA cosB - sinA sinB

To calculate trigonometric values for compound angle, we will switch to using the proofs with algebra of trigonometric functions, as it is simpler. But, equivalently a geometrical proof can be worked out.

Proof for sin(A-B) and cos(A-B) using previous results and algebra of trigonometric functions.

sin(A-B)
quad quad = sin(A+(-B))
quad quad = sinA cos(-B)+ cosA sin(-B)
quad quad = sinA cosB - cosA sinB

cos(A-B)
quad quad = cos(A+(-B))
quad quad = cosA cos(-B) - sinA sin(-B)
quad quad = cosA cosB + sinA sinB

Geometrical proof for sin(A-B) and cos(A-B) is outlined below. equate square of chord lengths to derive the result given.

bar(RT)^2 = bar(PQ1)^2
=> sin(A-B) = sinA cosB - cosA sinB

bar(RS)^2 = bar(PQ)^2
=> cos(A-B) = cosA cosB + sinA sinB

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

Compute sin75.

• cannot compute as 75^@ is not a standard angle
• consider 75^@ as sum of standard angles 45^@ and 30^@

The answer is 'consider 75^@ as sum of standard angles 45^@ and 30^@'.

sin75
quad = sin(45+30)
quad = sin45cos30 + cos45sin30
quad = (1+ sqrt(3))/(2sqrt(2))

Progress

Progress

It was geometrically proven that cos A+ B = cos A cos B minus sine A sine B. For the same result, we can work out a proof using algebra of trigonometric functions.
To calculate trigonometric values for compound angle, we will switch to using the proofs with algebra of trigonometric functions, as it is simpler. But, equivalently a geometrical proof can be worked out.
Proof for sine A minus B and cos A minus B using previous results and algebra of trigonometric funtions is given.
Geometrical proof for sine A minus B and cos A minus B is outlined below.
sine A minus B = sine A cos B minus cos A sine B;; cos A minus B = cos A cos B plus sine A sine B.
compute sine 75 degree.
cannot;not;compute
cannot compute as 75 degree is not a standard angle
consider;sum;45;30
consider 75 degree as sum of standard angles 45 degree and 30 degree
The answer is "consider 75 degree as sum of standard angles 45 degree and 30 degree"

we are not perfect yet...