Difference of Angles

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Difference of Angles

• Cosine and Sine
• The Proof
• Just sit back and enjoy the ride!

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Difference of Angles
a
?
Assume we are on the unit circle.
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Difference of Angles
a - ?
a
?
Assume we are on the unit circle.
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Difference of Angles
( cos (a ?) ,sin (a ?) )
( cos (a) ,sin (a) )
( cos (?) ,sin (?) )
a - ?
a
?
Assume we are on the unit circle.
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Difference of Angles
a (a ?) ?
( cos (a ?) ,sin (a ?) )
( cos (a) ,sin (a) )
( cos (?) ,sin (?) )
?
a - ?
a
?
Assume we are on the unit circle.
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Difference of Angles
( cos (a ?) ,sin (a ?) )
( cos (a) ,sin (a) )
( cos (?) ,sin (?) )
?
a - ?
a
?
(1,0)
Assume we are on the unit circle.
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Distance squared ( cos(a) cos(a ?) )2
(sin(a) - sin (a ?) )2
( cos (a ?) ,sin (a ?) )
( cos (a) ,sin (a) )
( cos (?) ,sin (?) )
?
a - ?
a
?
(1,0)
Distance squared (cos (?) 1)2 (sin (?) 0 )
2
They are the same!!
Assume we are on the unit circle.
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Set distances equal and simplify

• ( cos(a) cos(a ?) )2 (sin(a) - sin (a ?)
  )2 (cos (?) 1)2 (sin (?) 0 ) 2
• cos2(a) 2cos(a)(cos(a ?) cos2(a ?) sin
  2(a) - 2sin(a)sin(a ?) sin2(a ?)
• cos2(?) 2cos(?) 1 sin2(?)

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Set distances equal and simplify

• ( cos(a) cos(a ?) )2 (sin(a) - sin (a ?)
  )2 (cos (?) 1)2 (sin (?) 0 ) 2
• cos2(a) 2cos(a)(cos(a ?) cos2(a ?) sin
  2(a) - 2sin(a)sin(a ?) sin2(a ?)
• cos2(?) 2cos(?) 1 sin2(?)
• 1

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Set distances equal and simplify

• ( cos(a) cos(a ?) )2 (sin(a) - sin (a ?)
  )2 (cos (?) 1)2 (sin (?) 0 ) 2
• cos2(a) 2cos(a)(cos(a ?) cos2(a ?) sin
  2(a) - 2sin(a)sin(a ?) sin2(a ?)
• cos2(?) 2cos(?) 1 sin2(?)
• 1 1

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Set distances equal and simplify

• ( cos(a) cos(a ?) )2 (sin(a) - sin (a ?)
  )2 (cos (?) 1)2 (sin (?) 0 ) 2
• cos2(a) 2cos(a)(cos(a ?) cos2(a ?) sin
  2(a) - 2sin(a)sin(a ?) sin2(a ?)
• cos2(?) 2cos(?) 1 sin2(?)
• 1 1 2cos(a)(cos(a ?) - 2sin(a)sin(a ?)
  cos2(?) 2cos(?) 1 sin2(?)

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Set distances equal and simplify

• ( cos(a) cos(a ?) )2 (sin(a) - sin (a ?)
  )2 (cos (?) 1)2 (sin (?) 0 ) 2
• cos2(a) 2cos(a)(cos(a ?) cos2(a ?) sin
  2(a) - 2sin(a)sin(a ?) sin2(a ?)
• cos2(?) 2cos(?) 1 sin2(?)
• 1 1 2cos(a)(cos(a ?) - 2sin(a)sin(a ?)
  cos2(?) 2cos(?) 1 sin2(?)

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Set distances equal and simplify

• ( cos(a) cos(a ?) )2 (sin(a) - sin (a ?)
  )2 (cos (?) 1)2 (sin (?) 0 ) 2
• cos2(a) 2cos(a)(cos(a ?) cos2(a ?) sin
  2(a) - 2sin(a)sin(a ?) sin2(a ?)
• cos2(?) 2cos(?) 1 sin2(?)
• 1 1 2cos(a)(cos(a ?) - 2sin(a)sin(a ?)
  cos2(?) 2cos(?) 1 sin2(?)
• 1 1 2cos(a)(cos(a ?) - 2sin(a)sin(a ?)
  1

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Set distances equal and simplify

• ( cos(a) cos(a ?) )2 (sin(a) - sin (a ?)
  )2 (cos (?) 1)2 (sin (?) 0 ) 2
• cos2(a) 2cos(a)(cos(a ?) cos2(a ?) sin
  2(a) - 2sin(a)sin(a ?) sin2(a ?)
• cos2(?) 2cos(?) 1 sin2(?)
• 1 1 2cos(a)(cos(a ?) - 2sin(a)sin(a ?)
  cos2(?) 2cos(?) 1 sin2(?)
• 1 1 2cos(a)(cos(a ?) - 2sin(a)sin(a ?)
  1 2cos(?) 1

2 2cos(a)(cos(a ?) - 2sin(a)sin(a ?) 2
2cos(?) 2cos(a)(cos(a ?) - 2sin(a)sin(a
?) 2cos(?) cos(a)(cos(a ?)
sin(a)sin(a ?) cos(?)
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Formula for cosine
cos(a)(cos(a ?) sin(a)sin(a ?)
cos(?)
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Difference Formula for cosine

• cos(a)(cos(a ?) sin(a)sin(a ?) cos(?)
• If ß a ? and ? a ß
• And by substitution we get
• cos(a)cos(ß) sin(a)sin(ß) cos(a - ß)

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Difference Formula for cosine
cos(a)cos(ß) sin(a)sin(ß) cos(a - ß )

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Sum Formula for cosine

• cos(a)cos(ß) sin(a)sin(ß) cos(a - ß )
• cos(a)cos(-ß) sin(a)sin(-ß) cos(a (- ß) )
• cos(a)cos(ß) - sin(a)sin(ß) cos(a ß )

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Sum formula for sine

• cos(a)cos(ß) sin(a)sin(ß) cos(ß a)
• sin(x)cos(x-?/2)and sin(a-?/2)-cos(a)
• Now, we get
• sin(ß a) cos(ß a - ?/2 )

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Sum formula for sine

• cos(a)cos(ß) sin(a)sin(ß) cos(ß a)
• sin(a)cos (a-?/2)and sin(a-?/2)-cos(a)
• Now, we get
• sin(ß a) cos(ß (a - ?/2))

• By sum formula for cosine
• sin(ß a) cos(a ?/2)cos(ß) sin(a
  ?/2)sin(ß)
• sin(ß a) sin(a)cos(ß) cos(a)sin(ß)

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Sum formula for sine
sin(ß a) sin(a)cos(ß) cos(a)sin(ß)
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Difference formula for sine

• sin(ß a) sin(a)cos(ß) cos(a)sin(ß)
• sin(ß - a) sin(-a)cos(ß) cos(-a)sin(ß)
• sin(ß - a) - sin(a)cos(ß) cos(a)sin(ß)
• Because sine is an odd function
• And cosine is an even function
• So
• sin(ß - a) sin(ß )cos(a) - cos(ß)sin(a)

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Formulas

• cos(ß a) cos(a)cos(ß) sin(a)sin(ß)
• sin(ß a) sin(a)cos(ß) cos(a)sin(ß)
• sin(ß - a) sin(a)cos(ß) - cos(a)sin(ß)
• cos(ß - a) cos(a)cos(ß) sin(a)sin(ß)

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Double Angle

• cos(a a) cos(a)cos(a) sin(a)sin(a)
• cos(2a) cos2(a) sin2(a)
• sin(a a) sin(a)cos(a) cos(a)sin(a)
• sin(2a) 2sin(a)cos(a)

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Dont memorize them all

• Just know the double angle formulas
• cos(2x) is almost pythagorean thm
• cos(2x) cos2(x) sin2(x)
• If the angles are different then
• cos(xy) cos(x)cos(y) sin(x)sin(y)
• cos(x-y) cos(x)cos(-y) sin(x)sin(-y)
• Use cosine is even and sine is odd

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Dont memorize them all

• sin(2x) is twice the product
• sin(2x) 2sin(x)cos(x)
• If angles are different
• sin(xy) sin(x)cos(y) sin(y)cos(x)
• sin(x-y) sin(x)cos(-y) sin(-y)cos(x)
• Once again, use sine is odd, cosine is even

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