MAT 16A
Trigonometric functions
(A)
1
,
sin x
1
sec x =
,
cos x
sin x
cos x
cos x
cot x =
sin x
csc x =
tan x =
(B) Negative angle
sin(−x) = − sin x, cos(−x) = cos x
tan(−x) = − tan x, cot(−x) = − cot x
sec(−x) = sec x,
csc(−x) = − csc x
(C) Sum and difference formula
sin2 x + cos2 x = 1
sec2 x − tan2 x = 1
csc2 x − cot2 x = 1.
(D) Compound angle formula
sin(a + b) = sin a cos b + cos a sin b
cos(a + b) = cos a cos b − sin a sin b
tan a + tan b
tan(a + b) =
1 − tan a tan b
cot a cot b − 1
cot(a + b) =
cot a + cot b
sin(a − b) = sin a cos b − cos a sin b
cos(a − b) = cos a cos b + sin a sin b
tan a − tan b
tan(a − b) =
1 + tan a tan b
cot a cot b + 1
cot(a + b) =
cot b − cot a
(E) Double angle formula
sin 2a = 2 sin a cos a
cos 2a = cos2 a − sin2 a
= 1 − 2 sin2 a
= 2 cos2 a − 1
1
(F) Sum to product formula
sin(a + b) + sin(a − b) = 2 sin a cos b
cos(a + b) + cos(a − b) = 2 cos a cos b
sin(a + b) − sin(a − b) = 2 cos a sin b
cos(a + b) − cos(a − b) = −2 sin a sin b
(G) Interchange
π
π
− x = cos x,
sin
π2
tan
− x = cot x,
2π
csc
− x = sec x,
2
cos
− x = sin x
π2
cot
− x = tan x
2π
sec
− x = csc x
2
(H) Particular values
sin x
x=0
0
cos x
tan x
csc x
sec x
cot x
1
0
DNE
1
DNE
x=
1
√2
3
2
√1
3
2
√2
√3
3
π
6
π
4
x=
√1
2
√1
2
1
√
2
√
2
1
π
3
x√
=
3
2
1
√2
3
√2
3
2
√1
3
π
2
x=π
0
0
DNE
1
DNE
0
−1
0
DNE
−1
DNE
x=
1
(I) Derivatives
d
[sin x] = cos x,
dx
d
[tan x] = sec2 x,
dx
d
[sec x] = sec x tan x,
dx
2
d
[cos x]
dx
d
[cot x]
dx
d
[csc x]
dx
= − sin x
= − csc2 x
= − csc x cot x