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P. 416: 63-74 S
P. 416: 75-82 S
P. 416: 83-86 S
P. 416: 87-90
P. 416: 90-94
P. 416-7: 95-110 S
P. 417: 117, 118
P. 418: 123-126
Objective:
1. To use a variety of
trig formulas to find
exact trig values,
rewrite formulas,
solve equations, and
verify identities
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Assignment:
P. 416: 63-74 S
P. 416: 75-82 S
P. 416: 83-86 S
P. 416: 87-90
P. 416: 90-94
P. 416-7: 95-110 S
P. 417: 117, 118
P. 418: 123-126
There are a total of 19 new formulas in this section,
and by anyone’s count, that’s too many. When
you’re doing your assignment, make sure you
practice using all the formulas.
Try looking for certain patterns in the formulas to
help you remember them.
Remember that you’ll get to use a formula chart (or
note card) of your own construction on the test.
Remember also that you can earn a bonus of 10
points if you don’t use a note card.
Many of these formulas won’t get an extensive
workout (besides on your quizzes and tests)
until you take calculus. In calculus, you’ll use
them to simplify a trig function before you
integrate it.
sine
sin 2u  2 sin u cos u
cosine
cos 2u  cos 2 u  sin 2 u
 2 cos 2 u  1
 1 2 sin 2 u
tangent
2 tan u
tan 2u 
1  tan 2 u
These are
easy to
prove
using sum
formulas
and the
fact that
2u = u + u.
sine
1  cos 2u
sin u 
2
cosine
1  cos 2u
cos u 
2
tangent
1  cos 2u
tan u 
1  cos 2u
2
2
2
These are
easy to
prove if
you just
rearrange
cosine’s
double
angle
formulas.
sine
cosine
u
1  cos u
cos  
2
2
tangent
u
1  cos u
sin  
2
2
u
1  cos u 1  cos u
sin u
tan  


2
1  cos u
sin u
1  cos u
The sign of
each
quantity
depends
on the
quadrant
in which
u/2 lies.
1
sin u cos v  sin  u  v   sin  u  v  
2
1
cos u sin v  sin  u  v   sin  u  v  
2
1
cos u cos v  cosu  v   cosu  v 
2
1
sin u sin v  cosu  v   cosu  v 
2
These are
easy to
prove if
you add or
subtract
the sum
and
difference
formulas.
Prove the product-to-sum formula:
1
cos u cos v  cosu  v   cosu  v 
2
Rewrite sin(5x)cos(3x) as a sum or difference.
Remember when you were John Napier? You
took the product of two numbers and turned
it into a sum of two numbers using logarithms.
That was something folks did before there was
a calculator.
Before John Napier, [some rather insane] folks
did something similar with product-to-sum
formulas. That process had the rather
unwieldy name of prosthaphaeresis.
Use prosthaphaeresis to multiply 150 by 269.
uv
u v 
sin u  sin v  2sin 
 cos 

2
2




u v  u v 
sin u  sin v  2 cos 
 sin 

2
2

 

uv
u v 
cos u  cos v  2 cos 
 cos 

 2 
 2 
u v  u v 
cos u  cos v  2sin 
 sin 

2
2

 

These are easy to
prove using
the Productto-Sum
formulas
where your
angles are:
x
uv
2
and
y
uv
2
Prove the sum-to-product formula:
uv
u v 
sin u  sin v  2sin 
 cos 

2
2




Find the exact value of sin 195° + sin 105°.
Express cos(3x) – cos (4x) as the product of two trig
functions.
Solve for x.
sin5x  sin3x  0
Solve for x.
sin 4x  sin 2x  0
Verify the identity:
sin 6 x  sin 4 x
 tan 5 x
cos 6 x  cos 4 x
Objective:
1. To use a variety of
trig formulas to find
exact trig values,
rewrite formulas,
solve equations, and
verify identities
•
•
•
•
•
•
•
•
Assignment:
P. 416: 63-74 S
P. 416: 75-82 S
P. 416: 83-86 S
P. 416: 87-90
P. 416: 90-94
P. 416-7: 95-110 S
P. 417: 117, 118
P. 418: 123-126