ACT Math Formulas 26-27

Sine Periodic Shift (2π)

sin(θ + 2π) = sin(θ)

Cosine Periodic Shift (2π)

cos(θ + 2π) = cos(θ)

Tangent Periodic Shift (π)

tan(θ + π) = tan(θ)

Opposite Angle Identity for Sine

sin(−θ) = −sin(θ)

Opposite Angle Identity for Cosine

cos(−θ) = cos(θ)

Opposite Angle Identity for Tangent

tan(−θ) = −tan(θ)

Complementary Shift (π/2 − θ) for Sine

sin(π/2 − θ) = cos(θ)

Complementary Shift (π/2 − θ) for Cosine

cos(π/2 − θ) = sin(θ)

Complementary Shift (π/2 − θ) for Tangent

tan(π/2 − θ) = cot(θ)

Quarter-Turn Shift (θ + π/2) for Sine

sin(θ + π/2) = cos(θ)

Quarter-Turn Shift (θ + π/2) for Cosine

cos(θ + π/2) = −sin(θ)

Quarter-Turn Shift (θ + π/2) for Tangent

tan(θ + π/2) = −cot(θ)

Supplementary Shift (π − θ) for Sine

sin(π − θ) = sin(θ)

Supplementary Shift (π − θ) for Cosine

cos(π − θ) = −cos(θ)

Supplementary Shift (π − θ) for Tangent

tan(π − θ) = −tan(θ)

Half-Turn Shift (θ + π) for Sine

sin(θ + π) = −sin(θ)

Half-Turn Shift (θ + π) for Cosine

cos(θ + π) = −cos(θ)

Three-Quarter Turn Shift (θ + 3π/2) for Sine

sin(θ + 3π/2) = −cos(θ)

Three-Quarter Turn Shift (θ + 3π/2) for Cosine

cos(θ + 3π/2) = sin(θ)

tan(x)

sinx/cosx

secθ

1/cosθ

cscθ

1/sinθ

cotθ

1/tanθ

Law of Sines Formula

sin(A)/a = sin(B)/b = sin(C)/c

Unit Circle Coordinates at 0° (0 radians)

(cos 0, sin 0) = (1, 0)

Unit Circle Coordinates at 30° (π/6)

(cos 30°, sin 30°) = (√3/2, 1/2)

Unit Circle Coordinates at 45° (π/4)

(cos 45°, sin 45°) = (√2/2, √2/2)

Unit Circle Coordinates at 60° (π/3)

(cos 60°, sin 60°) = (1/2, √3/2)

Unit Circle Coordinates at 90° (π/2)

(cos 90°, sin 90°) = (0, 1)

Law of Sines

• a, b, c: Lengths of the sides of the triangle.

• A, B, C: Angles opposite to sides a, b, and c.

• R: Radius of the triangle's circumcircle (the circle that passes through all three vertices)

Law of Cosines

Standard Form of a Trig Function

y = Asin(B(x - C)) + D

• A (Amplitude): Controls the vertical stretch or height from the center line (|A|).

• B (Frequency factor): Used to find the period using the formula \(\frac{2\pi }{B}\) (or \(\frac{360^{\circ }}{B}\)).

• C (Phase shift): Horizontal shift to the left or right (note the minus sign in (x - C)).

• D (Vertical shift): Moves the graph up or down, which also sets the sinusoidal axis (midline

Two Functions of a Trig Function

y=sin(x) and y=3*sin(x)

Period (Trig Functions)

the distance between two peaks or two troughs of the curve in question

Amplitude (Trig Functions)

“A,” how far above and below the “midline” the curve goes

Phase Shift/Horizontal Shift (Trig Functions)

Caused by C and B (graph on calculator), changing C shifts whole graph to right or left, increasing B makes the period smaller (a compressed curve when graphed), and vice versa

Y Shift/Vertical Shift (Trig Functions)

Represented by D in the equation, if D has a value of 0, then the “midline” of the trig function will occur at y=0

y=sin(Bx)

y=cos(Bx)

y=tan(Bx)

Maximum and Minimum Values of Sinx and Cosx

Max: 1, Min: -1

Maximum and Minimum Values of Tanx, Cscx, Secx, and Cotx

No maximum, no minimum, (exept Cscx and Secx cannot equal 0)

Even Identity: Sine

sin(-x) = -sin(x)

Even Identity: Cosine

cos(-x) = cos(x)

Even Identity: Tangent

tan(-x) = -tan(x)

Even Identity: Cosecant

csc(-x) = -csc(x)

Even Identity: Secant

sec(-x) = sec(x)

Even Identity: Cotangent

cot(-x) = -cot(x)

Pythagorean Identity 1

sin²(x) + cos²(x) = 1

Pythagorean Identity 2

1 + tan²(x) = sec²(x)

Pythagorean Identity 3

1 + cot²(x) = csc²(x)

Sine Sum Identity

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

Sine Difference Identity

sin(a - b) = sin(a)cos(b) - cos(a)sin(b)

Cosine Sum Identity

cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

Cosine Difference Identity

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

Double-Angle Identity: Sine

sin(2x) = 2sin(x)cos(x)

Double-Angle Identity: Cosine

cos(2x) = cos²(x) - sin²(x)

Double-Angle Identity: Cosine Alternative 1

cos(2x) = 1 - 2sin²(x)

Double-Angle Identity: Cosine Alternative 2

cos(2x) = 2cos²(x) - 1

Double-Angle Identity: Tangent

tan(2x) = 2tan(x) / (1 - tan²(x))

Circle Formula

(h,k)=center, r=radius

Ellipse Formula (Horizontal)

Ellipse Formula (Vertical)

A and B on Ellipse

Hyperbola Formula (Horizontal)

Hyperbola Formula (Vertical)

Vertices of Horizontal Hyperbola

Vertices Vertical Hyperbola

Number in front of () in Hyperbola…

Tells how stretched or compressed (horizontally or vertically) a hyperbola is

A and B in Hyperbola

B: Draw a box, B is where diagonal asymptotes cross (height from center)

A: Distance from center of Hyperbola to two vertices

Combination

An arrangement where order does NOT matter (Key words: combinations, pairs, trios, groups)

Permutation

An arrangement where order DOES matter (Key words: unique, different, distinct, arrange, permutations, order)

Anatomy of Combinatoric Notation

n: total # of items

k: # of items chosen

Factorial Notation

Shown by an exclamation point (!), where x is an integer, is calculated by multiplying all integers between x and 1 together

Ex: 8! = 8×7×6×5×4×3×2×1

Factorial Term

(n+1)!=(n+1)*n!

Because of definition of Factorial → n!=n*(n-1)*n(n-2)*…

Intersection

∩ = where the sets overlap ONLY

Union

∪ = and/or, BOTH sets

(x,y) (Unit Circle)

(cosθ, sinθ)

Vertex of a Parabola

Difference of Squares

Trapezoid Area Formula

Circle Circumference and Area

Arc Length and Sector Area

Cylinder Volume

Sphere Volume and Surface Area

Cone Volume

Mean Formula