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)
