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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.
mathsAdvanced TrigonometryTrigonometric Identities for compound angles

### Compound Angles: Geometrical Proof for sin(A+B)

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

click on the content to continue..

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)
• (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))
• (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
• both the above

The answer is 'both the above'.
As 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)
• 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
• 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
• both the above

The answer is 'both the above'.

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

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

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