x, y, r - ReeTutorsMath.org

ReeTutorsMath.org - Trigonometry Formulas - Page 1 of 2
Double-Angle Formulas
Definitions
(OPPosite, ADJacent, HYPotenouse)
sin(2θ) = 2 sin(θ) cos(θ)
opp
sin(θ) =
hyp
hyp
csc(θ) =
opp
cos(2θ) = cos2 (θ) − sin2 (θ)
cos(2θ) = 2 cos2 (θ) − 1
cos(θ) =
adj
hyp
opp
tan(θ) =
adj
sec(θ) =
hyp
adj
adj
cot(θ) =
opp
Definitions - (x, y, r)
sin(θ) =
y
r
csc(θ) =
r
y
cos(θ) =
x
r
sec(θ) =
r
x
y
x
cot(θ) =
tan(θ) =
x
y
Changing To Sine And Cosine
tan(θ) =
sin(θ)
cos(θ)
sec(θ) =
1
cos(θ)
cot(θ) =
cos(θ)
sin(θ)
csc(θ) =
1
sin(θ)
cos(2θ) = 1 − 2 sin2 (θ)
tan(2θ) =
2 tan(θ)
1 − tan2 (θ)
Half-Angle Formulas
θ
=
sin
2
θ
cos
=
2
r
1 − cos(θ)
2
r
1 + cos(θ)
2
+
−
+
−
θ
1 − cos(θ)
tan
=
2
sin(θ)
θ
sin(θ)
tan
=
2
1 + cos(θ)
Sum and Difference Formulas
sin(A + B) = sin(A) cos(B) + cos(A) sin(B)
Pythagorean Identities
sin2 (θ) + cos2 (θ) = 1
sin(A − B) = sin(A) cos(B) − cos(A) sin(B)
cos(A + B) = cos(A) cos(B) − sin(A) sin(B)
cos(A − B) = cos(A) cos(B) + sin(A) sin(B)
tan2 (θ) + 1 = sec2 (θ)
1 + cot2 (θ) = csc2 (θ)
p
sin(θ) = +
1 − cos2 (θ)
−
p
cos(θ) = +
1 − sin2 (θ)
−
tan(A + B) =
tan(A) + tan(B)
1 − tan(A) tan(B)
tan(A − B) =
tan(A) − tan(B)
1 + tan(A) tan(B)
www.ReeTutorsMath.org/notes
ReeTutorsMath.org - Trigonometry Formulas - Page 2 of 2
Cofunction Formulas (in degrees)
Product-to-Sum Formulas
sin(A) sin(B) =
1
[cos(A − B) − cos(A + B)]
2
cos(A) cos(B) =
1
[cos(A − B) + cos(A + B)]
2
sin(A) cos(B) =
1
[sin(A − B) + sin(A + B)]
2
sin(θ)
cos(θ)
tan(θ)
cot(θ)
sec(θ)
csc(θ)
=
=
=
=
=
=
cos(90◦ − θ)
sin(90◦ − θ)
cot(90◦ − θ)
tan(90◦ − θ)
csc(90◦ − θ)
sec(90◦ − θ)
Cofunction Formulas (in radians)
Sum-to-Product Formulas
sin(A) + sin(B) = 2 sin
A+B
2
sin(A) − sin(B) = 2 cos
cos(A) + cos(B) = 2 cos
cos
A+B
2
A+B
2
cos(A) − cos(B) = −2 sin
sin
A+B
2
sin(θ) = cos
A−B
2
A−B
2
cos
A−B
2
sin
π
cos(θ) = sin
2
π
tan(θ) = cot
A−B
2
cot(θ) = tan
sec(θ) = csc
csc(θ) = sec
2
π
2
π
2
π
2
π
2
−θ
−θ
−θ
−θ
−θ
−θ
Power Reducing Formulas
sin2 (θ) =
Derivative Formulas (Calculus)
1 − cos(2θ)
2
1 + cos(2θ)
cos2 (θ) =
2
1 − cos(2θ)
tan2 (θ) =
1 + cos(2θ)
Even and Odd Trig Functions
sin(−θ)
csc(−θ)
tan(−θ)
cot(−θ)
cos(−θ)
sec(−θ)
=
=
=
=
=
=
− sin(θ)
− csc(θ)
− tan(θ)
− cot(θ)
cos(θ)
sec(θ)
sine is ODD
cosecant is ODD
tangent is ODD
cotangent is ODD
cosine is EVEN
secant is EVEN
d
sin(x)
dx
d
cos(x)
dx
d
tan(x)
dx
d
cot(x)
dx
d
sec(x)
dx
d
csc(x)
dx
www.ReeTutorsMath.org/notes
= cos(x)
= − sin(x)
= sec2 (x)
= − csc2 (x)
= sec(x) tan(x)
= − csc(x) cot(x)