Co­Function & Composite­Argument Properties
r
b
a
b/r
cos θ = a/r
tan θ = b/a
sec θ = r/a
csc θ = r/b
cot θ = a/b
sin θ =
sin (90° ­ θ) =
cos (90° ­ θ) =
tan (90° ­ θ) =
sec (90° ­ θ) =
csc (90° ­ θ) =
cot (90° ­ θ) =
CoFunction Properties
π
sin (90° ­ θ) = cos θ
sin ( ­ x) = cos x
2
tan (90° ­ θ) = cot θ
tan ( ­ x) = cot x
2
sec (90° ­ θ) = csc θ
sec ( ­ x) = csc x
2
cos (90° ­ θ) = sin θ
cos ( ­ x) = sin x
2
cot (90° ­ θ) = tan θ
cot ( ­ x) = tan x
2
csc (90° ­ θ) = sec θ
csc ( ­ x) = sec x
2
π
π
π
π
π
1
Co­Function & Composite­Argument Properties
Review Distance Formula
y2
(x2,y2)
d
y1 (x1,y1)
y2 ­ y1
x2 ­ x1
x2
x1
Distance formula
d2 = (x2 ­ x1)2 + (y2 ­ y1)2 Distance formula
d = (x2 ­ x1)2 + (y2 ­ y1)2 2
P
Q
P
Q
2
Co­Function & Composite­Argument Properties
(cos(A-B), sin(A-B))
P
Q
(1,0)
Since the 2 distances we just found are the same, set the two equal to each other...
solve for cos (A ­ B)
­2
­2
3
Co­Function & Composite­Argument Properties
cos ﴾A ­ B﴿ = cos A cos B + sin A sin B
Goal #2: Find a formula for cos (A + B)
Goal #3 - Find a formula for sin (A - B)
Cofunction Property:
sin θ =
_____________
sin (A - B) =
=
=
=
=
4
Co­Function & Composite­Argument Properties
sin ﴾A ­ B﴿ = sin A cos B ­ cos A sin B
Goal #4 - Find a formula for sin (A + B)
Goal #5: Find a formula for tan (A + B)
5
Co­Function & Composite­Argument Properties
THE LAST ONE! ...
Goal #6: Find a formula for tan (A - B)
tan (A - B) = tan (
)
=
Sum & Difference Formulas
cos (A ­ B) = cos A cos B + sin A sin B
cos (A + B) = cos A cos B ­ sin A sin B
sin (A ­ B) = sin A cos B ­ cos A sin B
sin (A + B) = sin A cos B + cos A sin B
tan A ­ tan B
tan (A ­ B) = 1 + tan A tan B
tan (A + B) = tan A + tan B
1 ­ tan A tan B
6
Co­Function & Composite­Argument Properties
Example 1: Find the exact value of each expression.
a) cos 75o
b) cos Example 1: Find the exact value of each expression.
c) sin
d) tan 15o 7
Co­Function & Composite­Argument Properties
Example 1: Find the exact value of each expression.
e) sin 80o cos 20o – cos 80o sin 20o
f) ,
,
YOU MUST FIRST DRAW A PICTURE FOR BOTH α AND β
Example 2: Find the exact value of each of the following under the given conditions:
(a)
(b)
(c)
(d)
8
Co­Function & Composite­Argument Properties
(a)
(b)
(c)
(d)
9
Attachments
OddOrEve.jpg
OddOrEve.jpg