Download Algebra II – Chapter 2 Lesson Plans

Advanced Algebra Trig Lesson Plans
Section 7.5 – Multiple-Angle and Product-to-Sum Formulas
Enduring Understandings: The student shall:
1. know and use double-angle formulas
2. be able to use power-reducing formulas
3. be able to use half-angle formulas to rewrite and evaluate functions
4. be able to use product-to-sum formulas
to rewrite and evaluate functions
Standards:
17 Trigonometry
Simplifies trigonometric expressions and solve trigonometric equations using
trigonometric identities
Essential Questions:
1. How do we deal with multiple angles?
Activities:
Double-Angle Formulas – MEMORIZE THE SINE AND COSINE DOUBLE ANGLE
FORMULAS and be able to use the rest.
Double Angle Formulas: Let both angles in the Sum and Difference Formulas equal
each other.
sin 2u = 2 sin u cos u
tan 2u 
2 tan u
1  tan 2 u
cos 2u = cos2 u – sin2 u = 2 cos2 u – 1 = 1 – 2 sin2 u
Power-Reducing Formulas: Solve the above for the squared function.
sin 2 u 
1  cos 2u
2
cos 2 u 
1  cos 2u
2
tan 2 
1  cos 2u
1  cos 2u
Half-Angle Formulas: Substitute v = u/2 in the above equations and then take square
root.
u 1  cos u
sin u
u
1  cos u
tan 


2
sin u
1  cos u
2
2
u
1  cos u
cos  
2
2
NOTE: the signs of sin(u/2) and cos(u/2) depend on the quadrant in which u/2 lies.
sin
Product-to-Sum Formulas: Easily verified from Sum and Difference Formulas in last
section.
1
1
sin u sin v  cosu  v   cosu  v 
cos u cos v  cosu  v   cosu  v 
2
2
1
1
sin u cos v  sin u  v   sin u  v 
cos u sin v  sin u  v   sin u  v 
2
2
Sum-to-Product Formulas: Let x = u + v, and y = u – v, solve for u and v, and substitute
into the product to sum formula.
  x  y   x  y 
sin x  sin y  2sin 
 cos

  2   2 
  x  y   x  y 
sin x  sin y  2cos
 sin 

  2   2 
  x  y   x  y 
cos x  cos y  2cos
 cos

  2   2 
  x  y   x  y 
cos x  cos y  2sin 
 sin 

  2   2 
Do: Day One: 4, 6, 10, 12, 20, 24, 32, 36, 46, 50, 60, 64,
Do: Day Two: 78, 80, 88, 92, 96, 114, 118, 120
Assessments:
Hw pg 570 - 573, # 1 – 73 by 4’s (19)
Hw pg 570 - 573, # 77 – 121 by 4’s (12)