Unit Circle
Inverse Trig Functions
Defiaitioa
y = sin-I x is equivalent to x
Y.f.(o,l)
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y = COS-I X is equivalent to x = cos y
0
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COS-I (cos (9)} = 9
sin (sin-I (x)) =x
sin-I (sin (9)) = 9
tan ( tan-I (x)) =x
tan-I (tan (9)) =8
Alteraate Notatioa
Function
Domain
Range
y=sin-' x
-ISxS 1
--~yS-
y=cos- I x
-ISxSl
O~ySft
y=tan- I x
-oo<x<oo
ft
ft
--<y<­
\(1.0)
ft
ft
2
2
2
• -I
•
sm x=arcsmx
COS-I X = arccos x
tan-I x = arctan x
2
Law of Sines, Cosines and Tangents
x
"i!r(,/3,_!)
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5.
4
(,/31)
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360°
T\
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180"
Iaverse Properties
cos ( COS-I (x)) = x
2' 2
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60°
6 150°
7.
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y =tan-I X is equivalent to x =tan y
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(./2 ./2)
3
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90°
2ft
3
I
(
,-...:2'2
2
sin y
240"
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2
Law ofSiaes
Law of TaageDts
sina = sinP = sinr
abc
a-b tant(a-p)
a+b = tant(a+ P)
Law of Cosmes
a Z = bl +c2 -2bccosa
b-c tant(P-r)
b+c = tanHP+r)
bl =az+cl-2accosp
For any ordered pair on the unit circle (x, y): cos 8 = x and sin 8 = y
C
Z
=a
l
+b2 -2abcosr
Example
cose;)=~
Sine;
)=
a-c
a+c
tanHa-r)
tanHa+r)
Mollweide's Formula
a+b = cost(a-p)
c
sintr
2
e 2005 Paul Dawkins
C 2005 Paul Dawkins
Formulas and Identities
Trig Cheat Sheet
Tangent and Cotangent Identities
sinO
0 cosO
tan O= - cot
Definition of the Trig Functions
cosO
Right triangle definition
Unit circ:Ie definition
Reciprocal Identities
o< 0 < !!.. or 0° < 0 < 90° .
For this definition 0 is any angle.
cscO = _1_
sinO
I
secO=-cosO
1
cot 0 = - tan 0
opposite I
I
-~,
'I
I .'
x
opposite
hypotenuse
adjacent
cosO
hypotenuse
tan 0 = opposite
adjacent
cscO
I
cosO=-secO
I
tan 0
cot 0
tan 2 0= l-cos(20)
l+cos(28)
Sum and Difference Formulas
±P)= sinacosp±cosa sinp
cos ( a ± P) = cos a cosp +sina sinp
sin( a
Pythagorean Identities
2
adjacent
. 0
sinO
sin 2 0 + cos 2 0 = I
0
sm =
sin 2 0:;:;: !(I-COS(20»)
2
1
2
cos 0 = -(I +cos(20»)
2
sinO
For this defmition we assume that
2
Half Angle Formulas
tan
2
tan 0 + I = sec 0
o
hypotenuse
opposite
o
hypotenuse
sec =
adjacent
adjacent
cot 0
opposite
csc =
Product to Sum Formulas
1+coe 0 == csc2 0
. 0 =-=y
y
sm
I
I
cscO=­
cosO=~=x
I
secO =­
x
sin (-0) =-sinO
csc(-B) =-cscO
I
cos (-8) =cosO
sec(-8) == secO
tanO=K
cotO=~
tan(-8)=-tanO
cot(-8)=-cote
x
y
y
sinasinp=~[cos(a-p)-cos(a+p)]
Even/Odd Formulas Periodic Formulas
Facts and Properties
If n is an integer.
sin(O+ 21rn) = sinO csc(O+ 21rn) = cscO
Domain
The domain is all the values of 0 that
can be plugged into the function.
sin 0,
cosO,
0 can be any angle
0 can be any angle
tanO,
O;t( n+~)1r' n=0,±I,±2,...
cscO,
O;tn1r, n=O,±I, ±2, ...
secO,
O;t(n+~)1r'
cotO,
O;tn1r, n=0,±I,±2,...
n=0,±I,±2,...
Range
The range is all possible values to get
out of the function.
-1~sinO~1
cscO:<!:1 andcscO~-1
-1~cosOSI
-QO
~
tan 0 S 00
secO:<!:landsecO~-1
-QO
S cot 0 ~oo
Period
The period of a function is the number,
T, such that J(O+T) J(O). SO, if w
is a fixed number and 0 is any angle we
have the following periods.
sin(wO) ~
T= 21r
cos(wO) ~
T= 21r
w
tan(wO) ~
T=!!..
w
T= 21r
w
T= 21r
w
W
csc(wO) ~
sec(wO) ~
cot(wO) ~
T=!!.. w
Ii:) 2005 Paul Dawkins
cos ( o+ 21rn) = cosO sec(O+ 21rn) == secO
tan (0 + 1rn):;:;: tan 0
tana
a+ ) = I+tanatanp
(-p
cot (0 + 1rn) = cote
Double Angle Formulas
cosacosp=~[ cos(a- p)+cos{a+p)]
sinacosp = i[sin(a+ p)+sin(a -
P)]
cosasinp= ![sin(a+ p)-sin(a -
P)]
2
Sum to Product Formulas
sina+Sin P =2sin( a;p)cos( a;p)
Sina-Sinp=2cos( a;p}in(a;p)
sin (20) = 2sinOcosO
cosa+cosp= 2COS( a; P)cos( a;p)
cos(28) = cos 2 0-sin 2 0
= 2cos
2
0-1
cosa-cosP=-2Sin(a;p}in( a;p)
= 1-2sin 2 0
Cofunction Formulas
tan (20) = 2 tan 0
l-tan 2 0
Degrees to Radians Formulas
If x is an angle in degrees and 1 is an
angle in radians then
1r
t
1rX
-==> 1
180 x
180
and
1801
x=-­
1r Sin(i-O)==COSO
cos (~-O) == sinO
csc(~-o) ==secO
sec( ~-O)
tan(~-O)=cotO
cot(~-O)=tane
== cscO
Ii:) 2005 Paul Dawkins