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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)


 »  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.


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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. sine A+B prooof 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)`.sine A+B prooof 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? sine A+B prooof 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)`.sine A+B proof 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)`.sine A+B proof 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`?sine A+B proof

  • `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`.sine A+B proof 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)`?sine A+B prooof 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.sine A+B prooof `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)`?sine A+B prooof 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.sine A+B prooof `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`sine A+B prooof sin(A+B) = sinAcosB + cosAsinB

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

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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.

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