Download Trig. Formulas

Formulas from Trigonometry
1
5
Radians and Degrees
π
− θ = cos θ
sin
2
If θ is an angle expressed in radians, and α is
the same angle measured in degrees, then
θ
α
=
.
π
180
tan
6
Definition of Trigonometric
Functions
sin θ =
opposite
hypotenuse
cos θ =
adjacent
hypotenuse
tan θ =
opposite
adjacent
cot θ =
adjacent
opposite
sec θ =
hypotenuse
adjacent
csc θ =
hypotenuse
opposite
π
− θ = cot θ
2
cot
π
− θ = tan θ
2
π
csc
− θ = sec θ
2
Symmetry, Periodicity
sin(−θ) = − sin θ
sin(θ + 2π) = sin θ
cos(θ + 2π) = cos θ
tan(θ + π) = tan θ
7
π
cos
− θ = sin θ
2
π
sec
− θ = csc θ
2
The length s of an arc on a circle of radius r
opposite the angle θ, measured in radians, is
given by the formula s = rθ .
2
Co-Functions
cos(−θ) = cos θ
sin(θ + π) = − sin θ
cos(θ + π) = − cos θ
tan(−θ) = − tan θ
Pythagorean Identities
sin2 θ + cos2 θ = 1
tan2 θ + 1 = sec2 θ
3
Values for Specific Angles
cot2 θ + 1 = csc2 θ
θ
0o
sin θ
cos θ
tan θ
0
0
1
0
30o
45o
60o
90o
π/6 √π/4 √π/3 π/2
3/2
1
√1/2 √2/2
2/2 1/2
0
√3/2
√
3/3
1
3 ±∞
8
Addition, Subtraction
sin(α + β) = sin α cos β + cos α sin β
sin(α − β) = sin α cos β − cos α sin β
4
Fundamental Identities
sin θ
cos θ
1
sec θ =
cos θ
1
cot θ =
tan θ
tan θ =
cos(α + β) = cos α cos β − sin α sin β
cos θ
sin θ
1
csc θ =
sin θ
cos(α − β) = cos α cos β + sin α sin β
cot θ =
1
tan(α + β) =
tan α + tan β
1 − tan α tan β
tan(α − β) =
tan α − tan β
1 + tan α tan β
9
π
2
arcsin(−x) = − arcsin x
arccos(−x) = π − arccos x
arctan(−x) = − arctan x
Double Angles, Half Angles
arcsin x + arccos x =
sin 2θ = 2 sin θ cos θ
cos 2θ = cos2 θ − sin2 θ
= 1 − 2 sin2 θ = 2 cos2 θ − 1
2 tan θ
tan 2θ =
1 − tan2 θ
1 − cos θ
1 + cos θ
2 θ
2 θ
sin
=
cos
=
2
2
2
2
θ
sin θ
1 − cos θ
tan
=
=
2
1 + cos θ
sin θ
10
13
sin α
sin β
sin γ
=
=
a
b
c
14
c2 = a2 + b2 − 2ab cos γ
15
1
A = ab sin γ
2
Heron’s Formula:
1
With s = (a + b + c) (= semiperimeter) the
2
area A of a triangle is given by
Sinusoids
A=
y = a sin(bx + c) + d
Amplitude
Period
Phase shift
Maximum value
Minimum value
Miscellaneous Formulas
Area A of a triangle:
Sinusoids are functions of the form
12
Law of Cosines
Products
1
sin α sin β = (cos(α − β) − cos(α + β))
2
1
cos α cos β = (cos(α − β) + cos(α + β))
2
1
sin α cos β = (sin(α − β) + sin(α + β))
2
11
Law of Sines
|a|
2π/|b|
−c/b
d + |a|
d − |a|
Inverse Functions
For −1 ≤ x ≤ 1 we define:
y = sin−1 x = arcsin x
if and only if
−π/2 ≤ y ≤ π/2 and sin y = x.
For −1 ≤ x ≤ 1 we define:
y = cos−1 x = arccos x
if and only if
0 ≤ y ≤ π and cos y = x.
For any real number x we define:
y = tan−1 x = arctan x
if and only if
−π/2 < y < π/2 and tan y = x
2
p
s(s − a)(s − b)(s − c)