Warmup Review 5.4
Per___ Name___________
1. Find the exact value of cos 285º showing
all steps.
1
Simplify (sin 5x)(cos 2x) – (cos 5x)(sin 2x)
2. Multiple choice
A sin7x
B cos7x
C sin3x
D cos3x
E 5sin2x
2
1
3. Simplify cos(5π/6 + x) +cos(5π/6 - x)
• Show all steps!
3
4. Given cos a =8/17
sin b = ½
3π/2 < a < 2π
π/2 < b < π
• Find sin(a – b)
• tan(a-b)
4
2
Trig/Precalc 5.5
Multiple-Angle and
Product-to-Sum Formulas
• Objectives:
• Use sums of angles to verify double angle
formulas.
• Use formulas to rewrite and evaluate trig
functions.
• Use formulas to rewrite real-life models.
5
5.4: Sum and Difference of
Angles Formulas
Look for patterns to help you memorize the formulas.
Create the double angle formulas by using u + u in the
formulas above.
6
3
Section 5.5: Double-Angle
Formulas
cos 2u has 3 forms because of
the Pythagorean identities.
7
HW Examples – Use the figure to find the
exact value of each trig function. “Exact”
means no decimals.
1
6. csc(2θ) =
θ
4
8. cot(2θ) =
8
4
HW Examples – Find the exact solutions of each
equation in the interval [0, 2π). “Exact” means no
decimals, it means to use the 30-45-60 trig ratios.
10. sin(2x) + cos x = 0
9
Find the exact values of sin 2u, cos 2u and
tan 2u using double angle formulas.
Q2
24. Cos u = -2/3
π/2 < u < π
Solution:
• Draw a triangle in Q2
• Find all 3 sides
• Solve for sin(2u), cos(2u), and
tan(2u).
sin 2u = 2sin u cos u
tan(2u ) =
2 tan u
1 − tan 2 u
cos 2u = 2cos2 u - 1
10
5
5.5: Half-Angle Formulas
11
Find the exact values using half angle
formulas.
Θ in Q1
36. sin θ/2 =
r
θ
1 − cos u
sin =
2
2
8
θ
15
12
6
Find the exact values using half angle
formulas.
1
Θ in Q1
38. sec θ/2 = cos(2θ )
r
8
θ
1 + cos u
cos =
2
2
θ
15
13
Hw 49-54 Find the exact values using half
angle formulas. Q3 π < u < 3π/2
-3
52. Cot u = 3
So tan u = 1/3
u
-1
u
=
2
u
cos =
2
u
tan =
2
sin
14
7
5.5: Product-to-Sum Formulas
* These formulas will be given to you on the test.
15
63 – 70 Use the product to sum formulas to
write the product as a sum or difference
64. 4 cos
π
3
sin
5π
6
cos u sin v =
1
sin ( u + v ) − sin ( u − v )
2
16
8
5.5: Sum-to-Product Formulas
* These formulas will be given to you on the test.
17
75 – 82 Use the sum-to-product formulas to
write the sum or difference as a product.
76. sin 3θ + sin θ
 u+v 
 u−v 
sin u + sin v = 2sin 
 cos 

2


 2 
18
9