Everything

D e fi n i t i o n of the Six TrigonometricFunctions
( 0 ,1 )
Riglr tt r ianglectefnition.r,*'here 0 < 0 < rr/2.
opposite
l ry p .
tan0:
Arljucerrt
2
Iryp.
sing=-g
csc6:
nyp.
opp.
adi.
hVg
=
cos 0
ODD,
-t-
aoJ,
r\ 2Lr v 5 /1
sec0=
cot 0 :
adj ,
adj .
opp.
rrt Jlt
\-f,T)
rrt 1r
\T,i)
T
r Jl, Jit
\-'ftf)
r ,ft l\
\--Ttl)
600
45'
300
(-1,0)
Circular function definitions, where 0 is any angle.
(1,0)
YT
330o
sin0=1 csc0:ry
cos0:tan0:1
XT
rx
VX
xy
scc0=cot0:-
r
\
rt
2 '
225'
L4 z4o"
lr
-1E1-, r
3150
(+,-L)
2/
rrt
e+,-*)
/\ - r2 , - f \z t
\Tt-TJ
---
rtt
r\ 2rr - / 25 r)
(0,_l)
Double-AngleFormulas
Reciprocalldentities
.11
slnu:
csc u
csc u
ll
s ln u
secu:-
cos u
s e cu
I
cot u
I
cotu:tun u
tanu:-
Tangentand Cotangentldentities
tatru:-
s in u
c os u
cosu
cOtu
s tn u
tilrt'u
Power-ReducingFormulas
l-cos2u
.1
sln'u -2
1
*
c
os 2u
cos." u = --;^ :
tan"
u
I * cotz u = csczu
Cofunctionldentities
. lrr
\
l r - u\ : s i n u
ttn\z - u) : cosu cost;
,|
\r,
/
lrr - \
l rr
\
.t.\t
,,) : secrl t t n ( t - u i = cot tl
/
lrr
\
ln
\
t..\t - u) - cscu c o t \ z - u
:tanu
)
NegativeAngle ldentities
sin(-u) : -sin u
csc(-u) : -csc u
s e c ( - u ) = s e cu
tirn2u:.-- I
2 c o s 2u - t - l ' - 2 s i n 2u
L
Pythagoreanldentities
sinZu*cos2u:l
I -f tan2u : sec2u
sin2u:2sinucosu
c o s2 u : c o s 2u - s i n 2u :
2tanu
cos(-u) = cosu
tan(-u) = -tan u
cot(-u) = -cot u
I -cos 2u
T+;;t
Sum-to-ProductForrnulas
sinu t sinv = 2',"(+)
."r(+)
\4/
\2/
sinu - siny = 2."'(+)''"(?)
cosu -r-cosv = z'"'(+)
."'(+)L /
\
\z/
cosu - cosv: -zt'"(+)
\
Z/
t,.(+)
\
L
Formulas
Product-to-Sum
I
s i n u s i n v : r l c o s ( u - v ) - cos(u-Fv)l
Surn and DifferenceFormulas
cosu cosv = jt.ort,r-v) * c o s ( u * v ) l
si n( ut v ) : s in u c o s v -t c o s u s i n v
c o s ( u t v ) : c o su c o sv T s i n u s i n v
sinucosv:jtrintu*v)t
tall(uiv)
:
tanu+tunv
;-=-_
lTtanutanv
si n(u- v)l
c o su s i n v - - j t r , n t u * u ) - s i n ( u- v ) l
/