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Thought-Process to Discover Knowledge

Welcome to **nub****trek**.

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.continue

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.continue

Welcome to **nub****trek**.

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. continue

The content is presented in small-focused learning units to enable you to

think,

figure-out, &

learn.

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.

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

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

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. continue

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. continue

*summary of this topic*

Voice

Voice

Home

sin(A+B)

» the coordinates of points

→ `P (cosA, sinA)`

→ `Q (cosB, sinB)`

→ `R (cos(A+B), sin(A+B))`

→ `Q1 (sinB, cosB)`

→ `T (0, 1)`

» equate the distance `bar(RT) = bar(PQ1)`

» `sin(A+B)``=sinAcosB ``+ cosAsinB`

cos(A+B)

» equate the distance `bar(RS) = bar(PQ2)`

» `cos(A+B)``=cosAcosB ``- sinAsinB`

*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

You are learning the free content, however do shake hands with a coffee to show appreciation.

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In this page, a simple and intuitive geometrical proof is explained for expressing `sin(A+B)` in terms of `sinA`, `sinB`, etc.

Starting on learning "Geometrical proof for sine A plus B ". ;; In this page, a simple and intuitive geometrical proof is explained for expressing sine A+B in terms of sine A, sine B, et cetera.

Consider points `P` and `Q` with angles `/_A` and `/_B` in unit circle as shown in the figure. What is the coordinate of point `P`?

- `(cosA, sinA)`
- cannot be computed with the given information

The answer is '`(cosA, sinA)`'.

And the point Q is `(cosB, sinB)`

In the given figure, point `R` is for angle `/_(A+B)`. What is the coordinate of point `R`?

- `(cos (A+B), sin(A+B))`
- cannot be computed with the given information

The answer is '`(cos (A+B), sin(A+B))`'.

In the given figure, coordinates of `P`, and `Q` are given. Can coordinate of `R` be calculated? In other words, given the trigonometric ratios of A and B , can we compute trigonometric ratios of `A+B`?

`sin(A+B) = ?`

`cos(A+B) = ?`

In the modified figure, point `Q1` is for angle `/_(90-B)`. What is the coordinate of point `Q1`?

- `(cos (90-B), sin(90-B))`
- `(sinB, cosB)`
- both the above

The answer is 'both the above'.

Since `cos(90-B) = sinB` and `sin(90-B) = cosB`.

In the given figure, consider point `T (0,1)`. What is the angle `/_ROT`?

- `90-(A+B)`
- cannot be computed

The answer is '`90-(A+B)`'.

`/_ROT`

`quad quad = /_SOT -/_SOR`

`quad quad = 90 - (A+B)`.

In the given figure, What is the angle `/_P O Q1`?

- `90-A-B`
- cannot be computed

The answer is '`90-A-B`'.

`/_POQ1`

`quad quad = /_SOT -/_SOP-/_Q1OT`

`quad quad = 90 - A - B`.

In the given figure:

`/_ROT = 90-(A+B)`

`/_POQ1= 90-A-B`. Which of the following is correct for chords `bar(RT)` and `bar(PQ1)`?

- angle subtended by the two chords are equal
- length of two chords are equal
- both the above

The answer is 'both the above'.

In the given figure, what is the length of chord `bar(RT)`? Note: coordinate of `R` is `(cos(A+B), sin(A+B))`. and coordinate of `T` is `(0,1)`.

- length of chord cannot be computed using the coordinates
- use the distance formula to compute the length of chords

The answer is 'use the distance formula'.

`bar(RT)^2`

`quad quad = (cos(A+B)-0)^2 + (sin(A+B)-1)^2`

`quad quad= 2-2sin(A+B)`

Square or length of chord `bar(RT)` is calculated as follows. `T(0,1)`

`R(cos(A+B), sin(A+B))`

`bar(RT)^2`

` quad quad = color(coral)((cos(A+B)-0)^2)`

` quad quad quad + color(deepskyblue)((sin(A+B)-1)^2)`

`quad quad = color(coral)(cos^2(A+B)) + color(deepskyblue)(sin^2(A+B))`

`quad quad quad + color(deepskyblue)(1-2sin(A+B))`

`quad quad= 2-2sin(A+B)`

In the given figure, what is the length of chord `bar(PQ1)`? Note: the coordinate of `P` is `(cosA, sinA)`. and the coordinate of `Q1` is `(sinB,cosB)`.

- length of chord cannot be computed using the coordinates
- use the distance formula to compute the length of chords

The answer is 'use the distance formula'.

`bar(PQ1)^2`

`quad quad = (cosA-sinB)^2 + (sinA-cosB)^2`

`quad quad = 2-2cosAsinB-2sinAcosB`

Square or length of chord `bar(PQ1)` is calculated as follows. `P(cosA,sinA)`

`Q1(sinB, cosB)`

`bar(PQ1)^2`

` quad quad = color(coral)((cosA-sinB)^2 `

` quad quad quad + color(deepskyblue)((sinA-cosB)^2)`

`quad quad = color(coral)(cos^2(A)) + color(deepskyblue)(sin^2(A))`

`quad quad quad +color(coral)(sin^2(B)) + color(deepskyblue)(cos^2(B))`

`quad quad quad color(coral)(-2 cosA sinB) color(deepskyblue)(-2sinA cosB)`

`quad quad= 2-color(coral)(2cosAsinB)-color(deepskyblue)(2sinAcosB)`

Equating the square of lengths of chords `bar(RT)` and `bar(PQ1)`.

`2-2sin(A+B)= 2-2cosAsinB-2sinAcosB` sin(A+B) = sinAcosB + cosAsinB

To compute the value of `cos(A+B)` the enclosed figure is used and the proof is outlined below. `P (cosA, sinA)`

`Q (cosB, sinB)`

`R (cos(A+B), sin(A+B))`

`S (1,0)`

`Q2 (cosB, -sinB)`

`bar(PQ2)^2 `

`quad quad = (cosA-cosB)^2 + (sinA+sinB)^2`

`quad quad = 2 -2cosAcosB+2sinAsinB`

`bar(RS)^2 `

`quad quad = (cos(A+B)-1)^2 + (sin(A+B)-0)^2`

`quad quad = 2-2cos(A+B)`

equating these two `cos(A+B) = cosAcosB - sinAsinB`

*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

*your progress details*

Progress

*About you*

Progress

Consider points P and Q with angles A and B as shown in the figure. What is the coordinate of point P?

cos;sine;A

cos A comma sine A

cannot;computed;with

cannot be computed with the given information

The answer is "cos A comma sine A" ;; And the point Q is cos B comma sine B.

In the given figure, point R is for angle A + B. What is the coordinate of point R?

cos;A;B;be;bee;sine

cos A + B comma sine A + B

cannot;computed;with

cannot be computed with the given information

The answer is "cos A+B comma sine A+B".

In the given figure, coordinates of P and Q are given. Can coordinate of R be calculated? In other words, given the trigonometric ratios of A and B can we compute trigonometric ratios of A plus B?

In the modified figure, point Q 1 is for angle 90 minus B. What is the coordinate of point Q1?

cos;minus

cos 90 minus B , comma sine 90 minus B

B;be;bee;sine

sine B comma cos B

both;above

both the above

The answer is "both the above". Since cos 90 minus B = sine B, and sine 90 minus B = cos B.

In the given figure, consider point T, 0 comma 1. What is the angle R O T?

90;minus;A;plus

90 minus, A plus B

cannot;computed

cannot be computed

The answer is "90 minus, A plus B"

In the given figure. What is the angle P O Q1?

90;minus;A

90 minus A, minus B

cannot;computed

cannot be computed

The answer is "90 minus A, minus B"

In the given figure, Angle R O T = 90 minus, A + B. and angle P O Q 1 = 90 minus A minus B. Which of the following is correct for chords R T and P Q 1?

angle;subtended;by

angle subtended by the two chords are equal

length;of

length of two chords are equal

both;above

both the above

The answer is 'both the above'.

In the given figure, what is the length of chord R T. Note that coordinate of R is cos A+B comma sine A+B. and Coordinate of T is 0 comma 1.

cannot;using

length of chord cannot be computed using the coordinates

distance;formula

use the distance formula to compute the length of chords

The answer is "use the distance formula" Proof is given in the next page.

Square or length of chord R T is calculated as follows. In this cos squared plus sine squared equals 1 formula is used.

In the given figure, what is the length of chord P Q 1? Note that the coordinate of point P is cos A, sine A. And the coordinate of point Q1 is sine B, cos B

cannot;using

length of chord cannot be computed using the coordinates

distance;formula

use the distance formula to compute the length of chords

The answer is "use the distance formula" The proof is given in the next page

Square or length of chord P Q1 is calculated as follows.

Equating the square of lengths of chords R T and P Q 1. We arrive at the result sine A + B = sine A cos B plus cos A sine B.

To compute the value of cos(A+B) the enclosed figure is used and the proof is outlined below. In the end we arrive at the result cos A+ B = cos A cosB minus sine A sine B.

sine A plus B = sine A cos B plus cos A sine B;; cos A + B = cos A cos B minus sine A sine B.