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Lone Star College-CyFair Formula Sheet
The following formulas are critical for success in the indicated course. Student CANNOT
bring these formulas on a formula sheet or card to tests and instructors MUST NOT
provide them during the test either on the board or on a handout. They MUST be
memorized.
Math 1314 College Algebra
FORMULAS/EQUATIONS
Distance Formula
If 𝑃
1
= (𝑥
1
, 𝑦
1
) and 𝑃
2
= (𝑥
2
, 𝑦
2
), the distance from 𝑃
1
to 𝑃
2
is
𝑑
( 𝑃
1
, 𝑃
2
)
√( 𝑥
2
− 𝑥
1
)
2
( 𝑦
2
− 𝑦
1
)
2
Standard Equation
Of a Circle
The standard equation of a circle of radius 𝑟 with center at (ℎ, 𝑘) is
(𝑥 − ℎ)
2
2
= 𝑟
2
Slope Formula
The slope 𝑚 of the line containing the points 𝑃
1
= (𝑥
1
, 𝑦
1
) and 𝑃
2
= (𝑥
2
, 𝑦
2
) is
𝑚 =
𝑦
2
− 𝑦
1
𝑥
2
− 𝑥
1
if 𝑥
1
≠ 𝑥
2
𝑚 is undefined if 𝑥
1
= 𝑥
2
Point-slope
Equation of a Line
The equation of a line with slope 𝑚 containing the points (𝑥
1
, 𝑦
1
) is
𝑦 − 𝑦
1
= 𝑚(𝑥 − 𝑥
1
)
Slope-Intercept Equation of
a Line
The equation of a line with slope 𝑚 and 𝑦-intercept 𝑏 is
𝑦 = 𝑚𝑥 + 𝑏
Quadratic Formula
The solutions of the equation 𝑎𝑥
2
𝑥 =
−𝑏 ± √𝑏
2
− 4 𝑎𝑐
2 𝑎
LIBRARY OF FUNCTIONS
Constant Function
Identity Function
Square Function
2
Cube Function
3
Reciprocal Function
Squared Reciprocal
Function
2
Square Root Function
Cube Root Function
3
Absolute Function
Exponential Function
𝑥
Natural Logarithm
Function
= ln 𝑥
Greatest Integer
Function
(𝟏, 𝒆)
(𝟎, 𝟏)
൬−𝟏,
𝟏
𝒆
൰
𝒚
𝒙
𝒇(𝒙) = 𝒆
𝒙
𝒇(𝒙) = 𝐥𝐧𝒙
(𝒆, 𝟏)
൬
𝟏
𝒆
, −𝟏൰
𝒙
𝒚
(𝟏, 𝟎)
𝒙
𝒚
𝒇(𝒙) = ⟦𝒙⟧
Math 1316 Trigonometry
Students in Trigonometry should know all the formulas from Math 1314 College Algebra
plus the following.
Unit Circle 𝒙
𝟐
𝟐
TRIGONOMETRIC FUNCTIONS
Of an Acute Angle
sin 𝜃 =
𝑏
𝑐
Opposite
Hypotenuse
csc 𝜃 =
𝑐
𝑏
Hypotenuse
Opposite
cos 𝜃 =
𝑎
𝑐
Adjacent
Hypotenuse
sec 𝜃 =
𝑐
𝑎
Hypotenuse
Adjacent
tan 𝜃 =
𝑏
𝑎
Opposite
Adjacent
cot 𝜃 =
𝑎
𝑏
Adjacent
Opposite
Of a General Angle
sin 𝜃 =
𝑏
𝑟
csc 𝜃 =
𝑟
𝑏
cos 𝜃 =
𝑎
𝑟
sec 𝜃 =
𝑟
𝑎
tan 𝜃 =
𝑏
𝑎
, 𝑎 ≠ 0 cot 𝜃 =
𝑎
𝑏
APPLICATIONS
Arc Length: 𝑠 = 𝑟𝜃, 𝜃 in radians
Area of Sector: 𝐴 =
1
2
2
𝜃, 𝜃 in radians
Angular Speed: 𝜔 =
𝜃
𝑡
, 𝜃 in radians
Linear Speed: 𝑣 =
𝑠
𝑡
SOLVING TRIANGLES
Law of Sine:
sin 𝐴
𝑎
sin 𝐵
𝑏
sin 𝐶
𝑐
Law of Cosine: 𝑎
2
2
2
− 2 𝑏𝑐 cos 𝐴
2
2
2
− 2 𝑎𝑐 cos 𝐴
2
2
2
− 2 𝑎𝑏 cos 𝐴
𝒓 =
√ 𝒂
𝟐
𝟐
(𝒂, 𝒃)
𝜽
𝒙
𝒚
𝑨
𝑩
𝑪
b
𝐜 𝒂
𝒔
LIBRARY OF TRIGONOMETRIC FUNCTIONS
Sine Function
= sin 𝑥
Cosine Function
= cos 𝑥
Tangent Function
𝑓(𝑥) = tan 𝑥
Secant Function
= sec 𝑥
Cosecant Function
= csc 𝑥
Cotangent Function
𝑓(𝑥) = cot 𝑥
Math 2412 Precalculus
Students in Precalculus should know all the formulas from Math 1314 College Algebra and
Math 1316 Trigonometry plus the following.
Half Angle Formulas
sin 𝜃 = ±√
1 −cos 2 𝜃
2
cos 𝜃 = ±√
1 +cos 2 𝜃
2
tan 𝜃 =
1 −cos 2 𝜃
sin 2 𝜃
Products and Quotients of Complex Numbers in Polar Form
Let 𝑧
1
1
(cos 𝜃
1
1
) and 𝑧
2
2
(cos 𝜃
2
2
Then 𝑧
1
2
1
2
[cos(𝜃
1
2
) + 𝑖 sin(𝜃
1
2
)]
and, if 𝑧
2
𝑧 1
𝑧 2
𝑟 1
𝑟 2
[
cos(𝜃
1
2
) + 𝑖 sin
1
2
)]
DeMoivre’s Theorem
If 𝑧 = 𝑟(cos 𝜃 + 𝑖 sin 𝜃) and n is a positive integer,
𝑛
𝑛
[cos
Complex Roots
Let 𝑤 = 𝑟(cos 𝜃
0
0
) be a complex number and let n ≥ 2 be an integer. If 𝑤 ≠ 0 ,
there are n distinct complex n th roots of 𝑤, given by the formula
𝑘
𝑛
[cos ൬
0
൰ + 𝑖 sin ൬
0
൰].
Where 𝑘 = 0 , 1 , 2 , ⋯ , 𝑛 − 1.
POLAR EQUATIONS OF CONICS(Focus at the Pole, Eccentricity 𝒆 )
Equation Description
1 − 𝑒 cos 𝜃
Directrix is perpendicular to the polar axis at a distance 𝑝 units to the left of
the pole: 𝑥 = −𝑝
1 + 𝑒 cos 𝜃
Directrix is perpendicular to the polar axis at a distance 𝑝 units to the right of
the pole: 𝑥 = 𝑝
1 + 𝑒 sin 𝜃
Directrix is parallel to the polar axis at a distance 𝑝 units above the pole:
1 − 𝑒 sin 𝜃
Directrix is parallel to the polar axis at a distance 𝑝 units below the pole:
Eccentricity
If 𝑒 = 1 , the conic is a parabola; the axis of symmetry is perpendicular to the directrix.
If 0 < 𝑒 < 1 , the conic is an ellipse; the major axis is perpendicular to the directrix.
If 𝑒 > 1 , the conic is a hyperbola; the transverse axis is perpendicular to the directrix.
ARITHMETIC SEQUENCE
𝑛
1
𝑛
1
1
1
+ 2 𝑑) + ⋯ + [𝑎
1
+ (𝑛 − 1 )𝑑]
𝑛
2
[ 2 𝑎
1
+ (𝑛 − 1 )𝑑]
𝑛
2
[𝑎
1
𝑛
]
GEOMETRIC SEQUENCE
𝑛
1
𝑛− 1
𝑛
1
1
1
2
1
𝑛− 1
𝑎
1
( 1 −𝑟
𝑛)
1 −𝑟
GEOMETRIC SERIES
If |𝑟| < 1 ,
1
1
1
2
1
𝑘− 1
∞
𝑘= 1
1
If
the infinite geometric series does not have a sum.
PERMUTATIONS/COMBINATIONS
C(n, r) = (
BINOMIAL THEOREM
𝑛
𝑛
𝑛− 1
2
𝑛− 2
𝑛− 1
𝑛
