SAT Math II

Amplitude (in terms of max & min)

(max + min) / 2

horizontal shift

(x = h)

vertical shift

(y = k)

Range of sin

-1>y<1 is where the shit goes down "less than or equal to"

0⁰

0 (1,0)

Basic Logarithmic Properties Involving One

1. logbb = 1 because 1 is the exponent to which b must be raised to obtain b. (b1 = b) 2. logb1 = 0 because 0 is the exponent to which b must be raised to obtain 1. (b0 = 1)

Sec=

1/x or the reciprocal of x (cos)

Csc=

1/y or the reciprocal of y(sin)

Half rotation

180° or pi

0° or 360°

2pi

Circumference

2pir

Period

2π/|b|

Full rotation

360° or 2pi

Cosine=

A/H, or x

y = A sin(Bx) + C

A: amplitude (vertical stretch/compression C: midline (vertical shift) B= 2π/ period (horizontal stretch/compression)

Area of a Sector

A=1/2(r^2)θ θ must be in radians

Complimentary angles

Add up to 90° if it is greater than pi/2, it has no compliment

Supplementary angles

Adds to 180°. If an angle is greater than pi, it has no supplement

The domain of cos

All real numbers

The domain of sin

All real numbers

Coterminal angles

Angles in the same position

Range of cos

Between -1>x< 1 shit be happening "equal to -1 & 1 as well"

Periodic function

Continues forever form, kind of like a wavelength looks. Like, copy and paste. Can be written as f(t+c) = f(t) where c is a real positive # sin(t+2πn)

Definition of the Logarithmic Function

Definition of the Logarithmic Function

Degree to radian

Degree • pi/180

Cos (even or odd?)

Even

The Change-of-Base Property

For any logarithmic bases a and b, and any positive number M, The logarithm of M with base b is equal to the logarithm of M with any new base divided by the logarithm of b with that new base.

Inverse Properties of Logarithms

For b > 0 and b 1, log base b, b∧x = x b∧log bx = x

Sin (and csc) is positive in quadrants?

I and II

Cos (and sec) is positive in quadrants?

I and IV

The Change-of-Base Property: Introducing Common and Natural Logarithms

Introducing Common Logarithms Introducing Natural Logarithms

The Power Rule

Let b and M be positive real numbers with b 1, and let p be any real number. The logarithm of a number with an exponent is the product of the exponent and the logarithm of that number.

The Product Rule

Let b, M, and N be positive real numbers with b 1. The logarithm of a product is the sum of the logarithms.

The Quotient Rule

Let b, M, and N be positive real numbers with b 1. The logarithm of a quotient is the difference of the logarithms.

Tangent

O/A or Y/x

Sine=

O/H, or y

Sine (even or odd?)

Odd

Tan and cot are (even or odd?)

Odd

90°

Pi/2

60°

Pi/3

45°

Pi/4

30°

Pi/6

If terminal rotates clockwise

Produces a negative angle

If terminal rotates counter clockwise

Produces a positive angle

Radian to degree

Radian measurement • 180/pi

Arc Length

S=rθ θ must be in radians

Sine Curve

Starts at 0

Cosine Curve

Starts at amplitude

The Domain of a Logarithmic Function

The domain of an exponential function of the form f(x) = b∧x includes all real numbers and its range is the set of positive real numbers. Because the logarithmic function reverses the domain and the range of the exponential function, the domain of a logarithmic function of the form: log base b, x :is the set of all positive real numbers. In general, the domain of: log base b, x: consists of all x for which g(x) > 0.

Common Logarithms

The logarithmic function with base 10 is called the common logarithmic function. f(x) = log₁₀ x The function f(x) = log x :is usually expressed

Natural Logarithms

The logarithmic function with base e is called the natural logarithmic function. f(x) = log base e, x The function f(x) = lnx is usually expressed

Radian

Used as a new way to measure angles in terms of pi.

Cot=

X/y reciprocal of tan y/x

Range

[-a, a]

a

amplitude

Phase Shift

bx+/-c=0 bx+/-c=2π

sin ø=?

cos (90-ø)

sin 72

cos 18

sin 10

cos 80

Even Functions When Neg.

cos(-θ)=cosθ sec(-θ)=secθ they're the same as the non-negated outcome so you should treat them as such

a is negative

horizontal flip on x-axis

b < 1

less cycles

b > 1

more cycles

b

period

increments

period / 4

Increments

period/4

secθ=

r/x, x≠0

cscθ=

r/y, y≠0

k is negative

shift down

k is positve

shift up

cos ø=?

sin (90-ø)

cos 70

sin 20

Odd Functions

sin(-θ)= -(sinθ) csc(-θ)= -(cscθ) tan(-θ)= -(tanθ) cot(-θ)= -(cotθ) the same as the non-negated but just negative.

reference angles

the acute angle formed by the terminal side of the angle and the x-axis

special right triangles

there are two special right triangles: 30-60-90 and 45-45-90

reference right triangles

triangles formed by drawing a vertical perpendicular line from the intersection of the terminal side of an angle and the unit circle to the x-axis Distances are always positive

b is negative

vertical flip across y-axis (in cosine, no visual change)

a < 1

vertical shrink

a > 1

vertical stretch

cosθ=

x/r

cotθ=

x/y, y≠0

Unit Circle

x^2+y^2=1 points on circle must satisfy equation

sinθ=

y/r

tanθ=

y/x, x≠0

90⁰

π/2 (0,1)

60⁰

π/3 (1/2, √3/2)

45⁰

π/4 (√2/2, √2/2)

30⁰

π/6 (√3/2, 1/2)

Domain

ℝ (-∞, ∞)

r=

√(x²+y²)