4.2 Review…Verify the identity:
1)
cot x ⋅ sin x = cos x
3) cot B +1
csc B
= cos B + sin B
2)
Mon Oct 31st Warm-up
cos A
sin A cot A
4) 1− cot 2 x
tan 2 x −1
=1
= cot x
2
Mon Oct 31st
9 HW Check. HW ?’s
9 Warm-up
Warm up
9 4.3: Sum, Difference, and
Cofunction Identities
9 Start HW …
4.3 Sum, Difference, and
R
Repeat
t Slid
Slide
Cofunction Identities
9How do you find the sin/cos/tan of an angle
th t iis nott iin standard
that
t d d position
iti or an angle
l that
th t iis
not a “special” angle?
(0,1)
(-1,0)
(1,0)
(0,-1)
9It’s a little complicated,
p
and has to do with the
pythagorean theroem (see pg. 235 for proof), but
we have formula’s to help…
The Sum and Difference Identities
( & y are angles
(x
l in degrees
d
or radians)
d
)
Repeat Slide
sin( x ± y ) = sin x cos y ± cos x sin y
cos( x ± y ) = cos x cos y ∓ sin x sin y
tan(( x ± y ) =
tan x ± tan y
1∓ tan x tan y
Simplify using an appropriate identity:
Repeat Slide
1) sin(x-π)
2) cos(x+π)
Strategies to finding exact values…
R
Repeat
t Slid
Slide
1) Change the given angle into a 30°/45°/60°
multiple
p if it isn’t one already.
y For example,
p
15° = 45° - 30°
7π/12 = π/3 + π/4
2) Next use the appropriate sum or difference
identity ~ expand it out & substitute known
values.
3) Simplify/multiply
Find the exact value of…(remember this
means do not use a calculator)
R
Repeat
t Slid
Slide
1) cos15°
2) tan75°
3) cos(x+y),
cos(x+y) given sinx = 3/5,
3/5 cosy = 4/5,
4/5 x is in Q2
Q2, and y is in Q1
Verify the identity…only
cos( x − y )
1) tan x + cot y =
cos x sin y
work on 1 side!
Repeat Slide
The Cofunction Identities
(
(replace
l
90° with
h π/2
/2 if
f x is in radian
d
measure))
Recall the cofunctions came from the
complementary angles in the right triangle…(pg.23)
90°-x
c
x
b
Repeat Slide
a
sinx = cos(90° – x)
tanx = cot(90
cot(90° - x)
secx = csc(90° - x)
cosx = sin(90° – x)
cotx = tan(90
tan(90° - x)
cscx = sec(90° - x)
Verify the identity using cofunction
identities for sine & cosine…
cosine only work on
side!
1) cot(π/2
t( /2 – x)) = ttanx
Repeat Slide
1
Wrap-up
• Use sum/difference formula’s when
trying
try
ng to f
find
nd ssin/cos/tan
n/cos/tan of angles
between 30°/45°/60° multiples.
•U
Use strategies
t t i (‘tricks’)
(‘t i k ’)
• Remember again,
g , be patient,
p
, don’t
erase everything….cross it out, start
over use previous work to help
over,
help…
• QUIZ Thurs/Fri!
• HW #28 p.241 #1-17 odd