Trigonometric Identities

Trigonometric Identities
Right-Triangle Definitions
sin(θ ) = Opp/Hyp
cos(θ ) = Adj/Hyp
tan(θ ) = Opp/Adj
cot(θ ) = Adj/Opp
sec(θ ) = Hyp/Adj
csc(θ ) = Hyp/Opp
Radians ⇐⇒ Degrees
xr =
πr
· t◦
180◦
t◦ =
180◦
· xr
πr
Reduction Formulas
sin(−x) = − sin(x)
π
sin
− x = cos(x)
π2
sin
+ x = cos(x)
2
sin(π − x) = sin(x)
cos(−x) = cos(x)
π
cos
− x = sin(x)
π2
cos
+ x = − sin(x)
2
cos(π − x) = − cos(x)
sin(π + x) = − sin(x)
cos(π + x) = − cos(x)
Pythagorean Theorem
sin2 (x) + cos2 (x) = 1
tan2 (x) + 1 = sec2 (x)
1 + cot2 (x) = csc2 (x)
Sum and Difference Formulas
sin(α + β ) = sin(α) cos(β ) + sin(β ) cos(α)
sin(α − β ) = sin(α) cos(β ) − sin(β ) cos(α)
cos(α + β ) = cos(α) cos(β ) − sin(β ) sin(α)
cos(α − β ) = cos(α) cos(β ) + sin(β ) sin(α)
tan(α + β ) =
tan(α) + tan(β )
1 − tan(α) tan(β )
tan(α − β ) =
tan(α) − tan(β )
1 + tan(α) tan(β )
Double Angle and Half Angle Formulas
r
1 − cos(θ )
2
1 + cos(θ )
cos(θ /2) = ±
2
1 − cos(θ )
sin(θ )
tan(θ /2) =
=
sin(θ )
1 − cos(θ )
sin(θ /2) = ±
r
sin(2θ ) = 2 sin(θ ) cos(θ )
cos(2θ ) = cos2 (θ ) − sin2 (θ )
tan(2θ ) =
2 tan(θ )
1 − tan2 (θ )
Euler’s Identity
eiθ = cos(θ ) + i sin(θ )
Derivative Formulas
d
sin(x) = cos(x)
dx
d
tan(x) = sec2 (x)
dx
d
sec(x) = sec(x) tan(x)
dx
d
arcsin(x) =
dx
d
arctan(x) =
dx
d
arcsec(x) =
dx
1
√
1 − x2
1
1 + x2
1
√
x x2 − 1
d
sinh(x) = cosh(x)
dx
d
tanh(x) = sech2 (x)
dx
d
sech(x) = sech(x) tanh(x)
dx
d
arcsinh(x) =
dx
d
arctanh(x) =
dx
d
arcsech(x) =
dx
1
√
1 + x2
1
1 − x2
1
√
x 1 − x2