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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.

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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. continue

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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. continue

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jogger provides the complete mathematical definition of the concepts.

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exercise provides practice problems to become fluent in the concepts.

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summary of this topic

Trigonometric Identities for compound angles

Trigonometric Identities for compound angles

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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. proof for sin(A-B) and cos(A-B) 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))`

your progress details

Progress

About you

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"

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