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Thought-Process to Discover Knowledge

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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.



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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)`
  • `(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))`
  • `(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
  • 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)`.sine A+B proof 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`?sine A+B proof

  • `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`.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
  • 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: 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.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
  • 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.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`

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

                            
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