9-2: Addition and
Subtraction Identities
Learning Goals:
•Use the addition and
subtraction identities for sine,
cosine & t...
Review
Find the exact values:
1. sin 2. cos180
2
3. sin 4. sin 270
3
5. cos 6. cos
2 2
°
°
π
π
π π
Review
60°
30°
1
23 45°
45°
1
1
2
Review
1
23 1
1
2
π
6
π
3
π
4
π
4
Review
Find the exact values:
1. sin 30 2. tan
6
3. cos 45 4. sin
3
5
5. cos 6. tan220
4
°
°
°
π
π
π
Does cos 40°+
cos 10°=cos 50°?
Analysis
Inquiring Minds Want To
Know…
cos(α+β)=cosα cosβ-sinα sinβ
cos(α-β)=cosα cosβ+sinα sinβ
Addition & Subtraction
Identities for Cosine:
Important Idea
cos...
Example
Find the exact value of
cos 75° using the values of
the function for 30° and
45°
Example
Find the exact value of
cos using the values
of the function for
and
π
3
π
4
7π
12
 
 ÷
 
Try This
Find the exact value of cos
15° using the values of the
function for 60° and 45°. Hint:
cos 15°=cos(60°-45°)
2 6
...
Try This
Find exact value of cos
using the values of the
function for and . Hint: cos
=
cos
2 6
4 4
+
π
12
 
 ÷
 
π
3...
Example
Rewrite the given
expression in terms of sin x
and/or cos x: cos(x + )π
sin(α+β)=sinα cosβ+cosα sinβ
sin(α-β)=sinα cosβ - cosα sinβ
Addition & Subtraction
Identities for Sine:
Important Idea
sin...
Example
Find the exact value of sin
75°. Simplify your answer.
Example
Find exact value of sin
Hint:
Simplify your answer.
13π
12
 
 ÷
 5π π 13π
4 6 12
 
− = ÷
 
Try This
Find exact value of sin
Hint:
5π
12
 
 ÷
 2π π 5π
3 4 12
 
− = ÷
 
Simplify your
answer.
6 2
4
+
Try This
Verify the identity:
sin(270°+x)=-cos x
Solution
sin(270 )
sin 270 cos cos270 sin
1(cos ) 0(sin )
cos
x
x x
x x
x
°+
= ° + °
= − +
= −
tan tan
tan( )
1 tan tan
α β
α β
α β
+
+ =
−
tan tan
tan( )
1 tan tan
α β
α β
α β
−
− =
+
Addition & Subtraction
Identitie...
tan tan
tan( )
1 tan tan
α β
α β
α β
±
± =
m
Addition & Subtraction
Identities for Tangent-the
condensed form:
Important I...
Example
Find the exact value of tan
15° using the values of the
tan function for 45° and 30°.
Simplify your answer.
Try This
If sin A= & cos B= ,
find tan(A+B).
4
5
3
5
24
7
−
Lesson Close
Describe a method for
finding the exact value
of sin 105°.
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Hprec9 2

  1. 1. 9-2: Addition and Subtraction Identities Learning Goals: •Use the addition and subtraction identities for sine, cosine & tangent ©2009 Roy L. Gover (www.mrgover.com)
  2. 2. Review Find the exact values: 1. sin 2. cos180 2 3. sin 4. sin 270 3 5. cos 6. cos 2 2 ° ° π π π π
  3. 3. Review 60° 30° 1 23 45° 45° 1 1 2
  4. 4. Review 1 23 1 1 2 π 6 π 3 π 4 π 4
  5. 5. Review Find the exact values: 1. sin 30 2. tan 6 3. cos 45 4. sin 3 5 5. cos 6. tan220 4 ° ° ° π π π
  6. 6. Does cos 40°+ cos 10°=cos 50°? Analysis Inquiring Minds Want To Know…
  7. 7. cos(α+β)=cosα cosβ-sinα sinβ cos(α-β)=cosα cosβ+sinα sinβ Addition & Subtraction Identities for Cosine: Important Idea cos( ) cos cos sin sin α β α β α β ± = m
  8. 8. Example Find the exact value of cos 75° using the values of the function for 30° and 45°
  9. 9. Example Find the exact value of cos using the values of the function for and π 3 π 4 7π 12    ÷  
  10. 10. Try This Find the exact value of cos 15° using the values of the function for 60° and 45°. Hint: cos 15°=cos(60°-45°) 2 6 4 4 +
  11. 11. Try This Find exact value of cos using the values of the function for and . Hint: cos = cos 2 6 4 4 + π 12    ÷   π 3 π 4π 12    ÷   π π 3 4   − ÷  
  12. 12. Example Rewrite the given expression in terms of sin x and/or cos x: cos(x + )π
  13. 13. sin(α+β)=sinα cosβ+cosα sinβ sin(α-β)=sinα cosβ - cosα sinβ Addition & Subtraction Identities for Sine: Important Idea sin( ) sin cos cos sin α β α β α β ± = ±
  14. 14. Example Find the exact value of sin 75°. Simplify your answer.
  15. 15. Example Find exact value of sin Hint: Simplify your answer. 13π 12    ÷  5π π 13π 4 6 12   − = ÷  
  16. 16. Try This Find exact value of sin Hint: 5π 12    ÷  2π π 5π 3 4 12   − = ÷   Simplify your answer. 6 2 4 +
  17. 17. Try This Verify the identity: sin(270°+x)=-cos x
  18. 18. Solution sin(270 ) sin 270 cos cos270 sin 1(cos ) 0(sin ) cos x x x x x x °+ = ° + ° = − + = −
  19. 19. tan tan tan( ) 1 tan tan α β α β α β + + = − tan tan tan( ) 1 tan tan α β α β α β − − = + Addition & Subtraction Identities for Tangent… Important Idea
  20. 20. tan tan tan( ) 1 tan tan α β α β α β ± ± = m Addition & Subtraction Identities for Tangent-the condensed form: Important Idea
  21. 21. Example Find the exact value of tan 15° using the values of the tan function for 45° and 30°. Simplify your answer.
  22. 22. Try This If sin A= & cos B= , find tan(A+B). 4 5 3 5 24 7 −
  23. 23. Lesson Close Describe a method for finding the exact value of sin 105°.
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