Transcript 5 5 Too Many Trig Formulas 01

Objective:
1. To use a variety of
trig formulas to find
exact trig values,
rewrite formulas,
solve equations, and
verify identities
•
•
•
•
•
•
•
•
•
Assignment:
P. 415: 1-8 S
P. 415: 9-18 S
P. 415: 19-22 S
P. 415: 23-28 S
P. 415: 19-34 S
P. 415: 35-40 S
P. 416: 41-48 S
P. 416: 49-54 S
P. 416: 59-64 S
You will be able to use a variety of trig
formulas to find exact trig values,
rewrite formulas, solve equations,
and verify identities
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.
Prove the double-angle formula:
sin 2u  2 sin u cos u
In the double-angle formulas, any double angle
will work: 2x, 4x, 6x, etc.
sin 6x  2sin3x cos3x
Since 6x = 2(3x)
cos8 x  2 cos 2 4 x  1
Since 8x = 2(4x)
Find a formula for cos 3u.
Solve for x.
2 cos x  sin 2x  0
Solve for x.
cos 2x  cos x  0
Verify the identity:
sec 2 x  sec 4 x
sec 2 x 
2  sec 2 x  sec 4 x
Use sin u = 3/5, 0 < u < π/2, to find sin 2u,
cos 2u, and tan 2u.
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.
Prove the power-reducing formula:
1  cos 2u
cos u 
2
2
Rewrite sin4 x as a sum of 1st powers of the cosines of
multiple angles.
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.
Prove the half-angle formula:
u
1  cos u
sin  
2
2
Find the exact value of cos (π/8)
Solve for x.
x
2  sin x  2 cos
2
2
2
Solve for x.
x
cos x  sin
2
2
2
Objective:
1. To use a variety of
trig formulas to find
exact trig values,
rewrite formulas,
solve equations, and
verify identities
•
•
•
•
•
•
•
•
•
Assignment:
P. 415: 1-8 S
P. 415: 9-18 S
P. 415: 19-22 S
P. 415: 23-28 S
P. 415: 19-34 S
P. 415: 35-40 S
P. 416: 41-48 S
P. 416: 49-54 S
P. 416: 59-64 S