Transcript Angles, Degrees, and Special Triangles

Multiple Angle & Product-to-Sum
Formulas
MATH 109 - Precalculus
S. Rook
Overview
• Section 5.5 in the textbook:
– Double-angle formulas
– Power-reducing formulas
– Half-angle formulas
– Product-to-sum & sum-to-product formulas
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Double-Angle Formulas
Double-Angle Formulas
• Often we are concerned with manipulating an angle
of the form ku where k is an integer
• We manipulate angles of the form 2u (called doubleangles) so often that we have derived formulas for
their usage:
– Can be derived from sum and difference formulas
– Formulas for angles with multiples other than 2 can
also be derived using the sum and difference formulas
cos 2u  cos2 u  sin 2 u
sin 2u  2 sin u cos u
2 tan u
tan 2u 
1  tan 2 u
 2 cos2 u  1
 1  2 sin 2 u
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Double-Angle Formulas (Example)
Ex 1: If
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sin x   , x  Q IV
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, find:
a) cos 2x
b) csc 2x
c) tan 2x
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Double-Angle Formulas (Example)
Ex 2: Use the double-angle formulas to find the
values of sin 120°, cos 120°, and tan 120°
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Power-Reducing Formulas
Power-Reducing Formulas
• Sometimes, we need to manipulate a trigonometric
function of the form sinnu, cosnu, tannu
• We do this by utilizing the power-reducing formulas
– Can be derived from the double-angle formulas
• These formulas allow us to write a trigonometric
function with power n into cosine functions with a
lower power
1  cos 2u
1  cos 2u
1  cos 2u
2
2
2
cos u 
tan u 
sin u 
2
1  cos 2u
2
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Power-Reducing Formulas
(Example)
Ex 3: Use the power-reducing formulas to
rewrite the expression in terms of the first
power of the cosine:
cos4 x
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Half-Angle Formulas
Half-Angle Formulas
• Sometimes we are concerned with manipulating an
angle of the form 1⁄2u called half-angles:
– Formulas can be derived from the power-reducing
formulas
u
u
sin
and
cos
– The sign for 2
2 is determined by which
quadrant u⁄2 terminates
u
1  cosu
u
1  cosu
cos  
sin  
2
2
2
2
u
sin u
tan 
2 1  cos u
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Half-Angle Formulas (Example)
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3
Ex 4: If csc u   ,   u 
, find:
3
2
a) sin u⁄2
b) cos u⁄2
c) tan u⁄2
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Product-to-Sum & Sum-toProduct Formulas
Product-to-Sum & Sum-to-Product
Formulas
• The preceding formulas can be used when we have
one angle
• However, situations arise where we wish to operate
on two DIFFERENT angles
– e.g. Products such as sin u cos v transform to sums
– e.g. Sums such as sin u + cos v transform to products
• When considering sines & cosines and two different
angles, we have four different situations that can
arise
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Product-to-Sum Formulas
• Product-to-Sum Formulas:
sin u sin v 
cosu cosv 
sin u cosv 
cosu sin v 
1
cosu  v   cosu  v 
2
1
cosu  v   cosu  v 
2
1
sin u  v   sin u  v 
2
1
sin u  v   sin u  v 
2
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Sum-to-Product Formulas
• Sum-to-Product Formulas:
u v u v
sin u  sin v  2 sin 
 cos

 2   2 
u v u v
sin u  sin v  2 cos
 sin 

 2   2 
u v u v
cosu  cos v  2 cos
 cos

 2   2 
u v uv
cosu  cos v  2 sin 
 sin 

 2   2 
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Product-to-Sum Formulas
(Example)
Ex 5: Use the product-to-sum formulas to write
the product as a sum or difference:
10 cos 75 cos 15
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Sum-to-Product Formulas
(Example)
Ex 6: Use the sum-to-product formulas to write
the sum or difference as a product:
sin 5  sin 3
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Summary
• After studying these slides, you should be able to:
– Use the double-angle, power-reducing, half-angle,
product-to-sum, and sum-to-product formulas to solve
problems
• Additional Practice
– See the list of suggested problems for 5.5
• Next lesson
– Solving Trigonometric Equations (Section 5.3)
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