Begin2.DVI

Trigonometry

Pythagorean identities

Using the Pythagorean theoremx^2 +y^2 =r^2 associated with a right triangle with

sides x, y and hypotenuse r, there results the following trigonometric identities,

known as the Pythagorean identities.
(x
r

) 2

+

(y

r

) 2

=1,

cos^2 θ+ sin^2 θ=1,

1 +

(y

x

) 2

=

(r

x

) 2

,

1 + tan^2 θ= sec^2 θ,

(

x

y

) 2

+ 1 =

(

r

y

) 2

,

cot^2 θ+ 1 = csc^2 θ,

Angle Addition and Difference Formulas

sin(A+B) = sinAcosB+ cosAsinB,

cos(A+B) = cosAcosB−sinAsinB,

tan(A+B) = 1 tan−tanA+ tanAtanBB,

sin(A−B) = sinAcosB−cosAsinB

cos(A−B) = cosAcosB+ sinAsinB

tan(A−B) =1 + tantanA−AtantanBB

Double angle formulas

sin2A=2 sinAcosA=

2 tanA

1 + tan^2 A

cos 2A= cos^2 A−sin^2 A= 1−2 sin^2 A= 2 cos^2 A−1 =^1 −tan

(^2) A
1 + tan^2 A
tan2A= 2 tanA
1 −tan^2 A
= 2 cotA
cot^2 A− 1
Half angle formulas
sin
A
2 =±
√
1 −cosA
2
cosA 2 =±
√
1 + cosA
2
tan
A
2 =±
√
1 −cosA
1 + cosA=
sinA
1 + cosA=
1 −cosA
sinA
The sign depends upon the quadrantA/ 2 lies in.
Multiple angle formulas
sin 3A=3 sinA−4 sin^3 A,
cos 3A=4 cos^3 A−3 cosA,
tan 3A=3 tanA−tan
(^3) A
1 −3 tan^2 A
,
sin4A=4 sinAcosA−8 sin^3 AcosA
cos 4A=8 cos^4 A−8 cos^2 A+ 1
tan4A= 4 tanA−4 tan
(^3) A
1 −6 tan^2 A+ tan^4 A