Co - function identities : Pythagorean identities : Periodicity identities : sin? e + cos? e =1 x+ 2n) = sinx = sin x COS tan? 0 + 1= sec? e cos(x +27) = cos x sin - x= COS X 2 1+ cot? e = csc? e tan(x+7) = tanx tan = cot x cot(x+7) = cot x Reciprocal identities : tan x cot 2 %3D csc5-) sec(x± 27) = sec x CSC X = CSC = sec x %3D sin x csc( x+27) = cSC X - X= CSC X sec x = sec COS X cot x = tan x Sum and difference formulas : sin(x+y) = sinxcosy ± cos x sin y cos(x ±y)= cos x cos y 7 sinx sin y Even - odd identities : sin(-x) = - sin x tan x ± tan y tan(x±y)= 17 tan x tan y cos(-x) = cos x tan(-x) = - tanx Product to sum formulas : Half - angle formulas : |- cos x sin x - siny = cos(x - y)- cos(x + sin X %3D 2 1+ cos x cos(x- y)+ cos(x + y)] COS X COs y = + CoS X Cos 2 (1- cos ) sin x - cos y =sin(x+y)+ sin(x – y)] tan sin x cOS X - siny = sin(x+y)- sin(x – y)] Sum to product: Law of sines : x ±y b. sinA sinB sinc sin x± siny = 2sin 2 COS X + y COS x + cos y = 2cos х-у Law of cosines : Cos %3! (*+y sin[) х-у a? = b² + c2 – 2bc cos A CoS x - cos y = -2 sin 2 b2 + c2 a? A = cos Double - angle formulas : 2bc Area of triangle: sin 20 = 2. sinecose cos 20 = cos?e – sin? e = 1-2sin? e =2cos?e –1 1 absinC = COS V5(5- a)( - (a+b+c) 2 tane (s-b)(s-c tan 20 %3D S-. 1- tan? e where s =
Co - function identities : Pythagorean identities : Periodicity identities : sin? e + cos? e =1 x+ 2n) = sinx = sin x COS tan? 0 + 1= sec? e cos(x +27) = cos x sin - x= COS X 2 1+ cot? e = csc? e tan(x+7) = tanx tan = cot x cot(x+7) = cot x Reciprocal identities : tan x cot 2 %3D csc5-) sec(x± 27) = sec x CSC X = CSC = sec x %3D sin x csc( x+27) = cSC X - X= CSC X sec x = sec COS X cot x = tan x Sum and difference formulas : sin(x+y) = sinxcosy ± cos x sin y cos(x ±y)= cos x cos y 7 sinx sin y Even - odd identities : sin(-x) = - sin x tan x ± tan y tan(x±y)= 17 tan x tan y cos(-x) = cos x tan(-x) = - tanx Product to sum formulas : Half - angle formulas : |- cos x sin x - siny = cos(x - y)- cos(x + sin X %3D 2 1+ cos x cos(x- y)+ cos(x + y)] COS X COs y = + CoS X Cos 2 (1- cos ) sin x - cos y =sin(x+y)+ sin(x – y)] tan sin x cOS X - siny = sin(x+y)- sin(x – y)] Sum to product: Law of sines : x ±y b. sinA sinB sinc sin x± siny = 2sin 2 COS X + y COS x + cos y = 2cos х-у Law of cosines : Cos %3! (*+y sin[) х-у a? = b² + c2 – 2bc cos A CoS x - cos y = -2 sin 2 b2 + c2 a? A = cos Double - angle formulas : 2bc Area of triangle: sin 20 = 2. sinecose cos 20 = cos?e – sin? e = 1-2sin? e =2cos?e –1 1 absinC = COS V5(5- a)( - (a+b+c) 2 tane (s-b)(s-c tan 20 %3D S-. 1- tan? e where s =
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.3: The Addition And Subtraction Formulas
Problem 78E
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Hello,
When I asked an earlier question, the response regarded a identity:
1-cos(x) = 2sin2(x/2)
I do not understand where this came from. Does this stem from another commonly known trig identity like the ones shown in the attached photo?
![Co - function identities :
Pythagorean identities :
Periodicity identities :
sin? e + cos? e =1
x+ 2n) = sinx
= sin x
COS
tan? 0 + 1= sec? e
cos(x +27) = cos x
sin
- x= COS X
2
1+ cot? e = csc? e
tan(x+7) = tanx
tan
= cot x
cot(x+7) = cot x
Reciprocal identities :
tan x
cot
2
%3D
csc5-)
sec(x± 27) = sec x
CSC X =
CSC
= sec x
%3D
sin x
csc( x+27) = cSC X
- X= CSC X
sec x =
sec
COS X
cot x =
tan x
Sum and difference formulas :
sin(x+y) = sinxcosy ± cos x sin y
cos(x ±y)= cos x cos y 7 sinx sin y
Even - odd identities :
sin(-x) = - sin x
tan x ± tan y
tan(x±y)= 17 tan x tan y
cos(-x) = cos x
tan(-x) = - tanx
Product to sum formulas :
Half - angle formulas :
|- cos x
sin x - siny =
cos(x - y)- cos(x +
sin X
%3D
2
1+ cos x
cos(x- y)+ cos(x + y)]
COS X COs y =
+ CoS X
Cos
2
(1- cos )
sin x - cos y =sin(x+y)+ sin(x – y)]
tan
sin x
cOS X - siny = sin(x+y)- sin(x – y)]
Sum to product:
Law of sines :
x ±y
b.
sinA sinB sinc
sin x± siny = 2sin
2
COS
X + y
COS x + cos y = 2cos
х-у
Law of cosines :
Cos
%3!
(*+y sin[)
х-у
a? = b² + c2 – 2bc cos A
CoS x - cos y =
-2 sin
2
b2 + c2
a?
A = cos
Double - angle formulas :
2bc
Area of triangle:
sin 20 = 2. sinecose
cos 20 = cos?e – sin? e = 1-2sin? e =2cos?e –1
1 absinC
= COS
V5(5- a)( -
(a+b+c)
2 tane
(s-b)(s-c
tan 20
%3D
S-.
1- tan? e
where s =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa9d887de-c42e-4811-95c0-31fa01032135%2F414c0bdb-5ab1-4156-bbe0-16190e50d61f%2F4cfozd3.png&w=3840&q=75)
Transcribed Image Text:Co - function identities :
Pythagorean identities :
Periodicity identities :
sin? e + cos? e =1
x+ 2n) = sinx
= sin x
COS
tan? 0 + 1= sec? e
cos(x +27) = cos x
sin
- x= COS X
2
1+ cot? e = csc? e
tan(x+7) = tanx
tan
= cot x
cot(x+7) = cot x
Reciprocal identities :
tan x
cot
2
%3D
csc5-)
sec(x± 27) = sec x
CSC X =
CSC
= sec x
%3D
sin x
csc( x+27) = cSC X
- X= CSC X
sec x =
sec
COS X
cot x =
tan x
Sum and difference formulas :
sin(x+y) = sinxcosy ± cos x sin y
cos(x ±y)= cos x cos y 7 sinx sin y
Even - odd identities :
sin(-x) = - sin x
tan x ± tan y
tan(x±y)= 17 tan x tan y
cos(-x) = cos x
tan(-x) = - tanx
Product to sum formulas :
Half - angle formulas :
|- cos x
sin x - siny =
cos(x - y)- cos(x +
sin X
%3D
2
1+ cos x
cos(x- y)+ cos(x + y)]
COS X COs y =
+ CoS X
Cos
2
(1- cos )
sin x - cos y =sin(x+y)+ sin(x – y)]
tan
sin x
cOS X - siny = sin(x+y)- sin(x – y)]
Sum to product:
Law of sines :
x ±y
b.
sinA sinB sinc
sin x± siny = 2sin
2
COS
X + y
COS x + cos y = 2cos
х-у
Law of cosines :
Cos
%3!
(*+y sin[)
х-у
a? = b² + c2 – 2bc cos A
CoS x - cos y =
-2 sin
2
b2 + c2
a?
A = cos
Double - angle formulas :
2bc
Area of triangle:
sin 20 = 2. sinecose
cos 20 = cos?e – sin? e = 1-2sin? e =2cos?e –1
1 absinC
= COS
V5(5- a)( -
(a+b+c)
2 tane
(s-b)(s-c
tan 20
%3D
S-.
1- tan? e
where s =
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