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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

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Read in the blogs more about the unique learning experience at nubtrek.

User Guide

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

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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

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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

### Compound Angles: Geometrical Proof for sin(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 and intuitive geometrical proof is explained for expressing sin(A+B) in terms of sinA, sinB, etc.

Keep tapping on the content to continue learning.
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.

we are not perfect yet...