MSTE Merged

Quadratic Formula

Sum of Roots of a Quadratic Equation

Product of Roots of a Quadratic Equation

Binomial Expansion

r^th term of a Binomial Expansion

r^th = nCm * a^n-m * b^m

Arithmetic Progression: Common Difference

d = a₂ - a₁

Arithmetic Progression: n^th term

a_n = a₁ + (n-1)d

Arithmetic Progression: Sum of first n terms

Geometric Progression: Common Ratio

r = a₂ / a₁

Geometric Progression: n^th term

a_n = a₁ * r^n-1

Geometric Progression: Sum of first n terms

Sum of Infinite Geometric Progression

S_n = a₁ / (1 - r)

Harmonic Progression: Common Difference of Reciprocals

d = 1 / a₂ - 1 / a₁

Harmonic Progression: n^th Term

a_n = 1 / (a₁ + (n - 1)d)

Arithmetic Mean

Geometric Mean

Harmonic Mean

Relationship between Means of Two Numbers

AM * HM = GM²

Types of Variation and their Relationships

Direct: x = ky

Indirect: x = k / y

Joint: x = kyz

Permutation: General Formula

nPr = n! / (n - r)!

Permutation: The number of permutations of n objects taken

all at a time in which there are alike objects

n! / (n₁! n₂! n₃! …)

Permutation: The number of ways of partitioning a set of n

objects into r groups with specific number of elements per group.

n! / (n₁! n₂! n₃! …)

Cyclic Permutation: taken r at a time

(nCr) (r - 1)! = nPr / r

Cyclic Permutation: taken all at a time

(n - 1)!

Combination: General Formula

nCr = n! / [r! (n - r)!]

Summation of Combinations

Probability: Multiplication Rule

P(A and B) = P(A) * P(B | A)

Note: P(B | A) = Probability that event B occurs given that event A occurred

Probability: Bayes’ Theorem

P(A) * P(B | A) = P(B) * P(A | B)

Probability: AND Rule for independent events

P(A and B) = P(A) * P(B)

Probability: Addition Rule

P(A or B) = P(A) + P(B) - P(A and B)

Probability: Complementary Events

P(A) = 1 - P(A^c)

Note: A^c = event not A

Binomial Probability Distribution

P(x) = p^x * q^{n - x} * nCx

Geometric Probability Distribution

P(n) - q^n-1 * p

Poisson Probability Distribution

Standard Normal Distribution (CALCTECH)

1. Get value of Z-score

2. Stat Mode → 1 then AC (don’t type anything)

3. Apps → 7: Distr

   1. For 1: P( → Input the Z-score, you will get the area to the left of the z-score to negative infinity.

   2. For 2: Q( → Input the Z-score, you will get the area between the z-score and the middle point (mean of the dataset).

   3. For 3: R( → Input the Z-score, you will get the area to the right of the z-score to positive infinity.

Pythagorean Theorem

a² + b² = c²

Oblique Triangle: Sine Law

a / sinA = b / sinB = c / sinC

Oblique Triangle: Cosine Law

a² = b² + c² - 2bc cosA

b² = a² + c² - 2ac cosB

c² = a² + b² - 2ab cosC

Trigonometry: Pythagorean Relations

sin²A + cos²A = 1

1 + tan²A = sec²A

1 + cot²A = csc²A

Sum and Difference of Two Angles: Sine

sin(A + B) = sinA cosB + cosA sinB

sin(A - B) = sinA cosB - cosA sinB

Sum and Difference of Two Angles: Cosine

cos (A + B) = cosA cosB - sinA sinB

cos (A + B) = cosA cosB + sinA sinB

Sum and Difference of Two Angles: Tangent

tan (A + B) = tan A + tan B / (1 - tanA tan B)

tan (A - B) = tan A - tan B / (1 + tanA tan B)

Double Angle: Sine

sin2A = 2 sinA cosA

Double Angle: Cosine

cos 2A = cos²A - sin²A

Double Angle: Tangent

tan 2A = 2 tanA / (1 - tan²A)

Half- Angle Identities

Square Identites

Area of Triangle: Base and Altitude

A = ½ bh

Area of Triangle: Two sides and included angle

A = ½ ab sinC

A = ½ bc sinA

A = ½ ac sinB

Area of Triangle: Three sides

Heron’s Formula: A = √s(s - a)(s - b)(s - c)

s = ½ (a + b + c)

Area of Triangle: Three angles and one side

A = a² sinB sinC / 2 sinA

A = b² sinA sinC / 2 sinB

A = c² sinA sinB / 2 sinC

Period and Amplitude: Sine Function

y = A sin(Bx + C) + D

Period = P = 2π / B

Amplitude = A

Period and Frequency: Cosine Function

y = A cos(Bx + C) + D

Period = P = 2π / B

Amplitude = A

Period and Frequency: Tangent Function

y = A tan(Bx + C) + D

Period = P = π / B

Medians of a Triangle

m_a = ½ √ 2b² + 2c² - a²

m_b = ½ √ 2a² + 2c² - b²

m_c = ½ √ 2a² + 2b² - c²

Altitudes of a Triangle

h_a = 2A_T / a

h_b = 2A_T / b

h_c = 2A_T / c

Angle Bisectors of a Triangle

b_a = 2√bcs(s - a) / (b + c)

b_b = 2√acs(s - b) / (a + c)

b_c = 2√abs(s - c) / (a + b)

Spherical Excess

E = A + B + C - 180°

Spherical Deflect

D = 360° - (a + b + c)

Right Spherical Triangle

Sin-Ta-Ad: Sine of middle = product of tangent of adjacent parts

i.e., sinb = tana * tan(co-A)

Sin-Co-Op: Sine of middle = product of cosine of opposite parts

i.e., sin(co-c) = cosa * cosb

Note: co-A = complement of A (equal to 90 - A)

Oblique Spherical Triangle: Sine Law

sina / sinA = sinb / sinB = sinc / sinC

Oblique Spherical Triangle: Cosine Law for Sides

cosa = cosb cosc + sinb sin c cosA

cosb = cosa cosc + sina sin c cosB

cosc = cosa cosb + sina sin b cosC

Oblique Spherical Triangle: Cosine Law for Angles

cosA = -cosB cosC + sinB sinC cosa

cosB = -cosA cosC + sinA sinC cosb

cosC = -cosA cosB + sinA sinB cosc

Area of Spherical Triangle

A = π r² E / 180°

Note: See pic for computation of spherical excess, given the sides

Conversion: minute of arc to feet

1 minute of arc = 6080 ft

Conversion: statute mile to feet

1 statute mile = 5280 ft

Conversion: nautical mile to statute mile

1 nautical mile = 1.1516 statute mile

Conversion: knot to nautical mile per hour

1 knot = 1 nautical mile per hour

Polygons: Sum of Interior Angles

Σθ = 180° (n - 2)

n = number of sides

Polygons: Number of Diagonals

d = n(n - 3) / 2

Perimeter of Regular Polygon

P = sn

Area of Regular Polygon

A = Pa / 2 = s² n / (4 tan(180° / n))

a = apothem

Area of Parallelogram

A = bh

Area of Trapezoid

A = h(b₁ + b₂) / 2

Area of Kite and Rhombus

A = d₁d₂ / 2

Area of General Quadrilateral

A = √(s - a)(s - b)(s - c)(s - d) - abcd cos²θ

θ = (A + C) / 2 or θ = (B + D) / 2

s = (a + b + c + d) / 2

Circle: Circumference

C = πd = 2πr

Circle: Area

A = πr²

Arc Length

s = rθ

Note: θ is in radians.

Circle: Sector Area:

A = r²θ / 2 = πr²θ / 360

Circle: Segment Area

A = r² (θ - sinθ) / 2

Circle: Two Chords Rule

a × b = c × d

Circle: Two Secants Rule

AC × AB = AE × AD

Circle: Secant-Tangent Rule

AB² = AC × AD

Cyclic Quadrilateral

d₁ × d₂ = ac + bd

Circle Inscribed in a Triangle

A_T = rs

s = ½ (a + b + c)

Circle Circumscribing a Triangle

A_T = abc / 4R

Escribed Circle

A_T = r(s - a)

Prism: Volume and Surface Area

V = A_B h

TSA = 2A_B + LSA

Truncated Prism: Volume and Surface Area

V = A_B h_ave

TSA = A₁ + A₂ + LSA

Pyramid: Volume and Surface Area

V = A_B h / 3

TSA = A_B + LSA

Tetrahedron: Volume and Surface Area

V = a³ / 6√2

SA = a² √3

Hexahedron (Cube): Volume and Surface Area

V = a³

SA = 6a²

Octahedron: Volume and Surface Area

V = √2 / 3 × a³

SA = a² × 2√3

Cylinder: Volume and Surface Area

V = A_B h = π r²

TSA = 2π r² + 2π r h

Cone: Volume and Surface Area

V = π r² h / 3

TSA = π r² + π r l

Sphere: Volume and Surface Area

V = 4π r³ / 3

SA = 4π r²

Volume of a Spherical Segment

1 Base: V = πh²(3R - h) / 3

2 Bases: V = πh(3a² + 3b² + h²) / 6