Formulas - Evaluation 4 Flashcards
Radian/Degree conversions
Radian to degrees:
x radian * 180/π
Degrees to radian:
xº * π/180
Arc length
S = rθ
S: arc length
r: radius
θ: angle in rad
Area of a sector
A = 1/2r²θ
A: area of sector
r: radius
θ: angle in rad
Trig function
cosθ = A/H
sinθ = O/H
tanθ = O/A (sinθ/cosθ)
cotθ = 1/tanθ
secθ = 1/cosθ
cosecθ = 1/sinθ
Unit circle
Radius -1
S = rθ
P(x,y)
x = cosθ
y = sinθ
Quadrantal Angles
Q1: 0 - π/2
Q2: π/2 - π
Q3: π - 3π/2
Q4: 3π/2 - 2π
Finding the 6-trig functions
find the lengths of all sides of the triangle with pythagoras theorem then calculate the 6 trig functions based on the knowledge that:
x = cosθ
y = sinθ
a² + b² = 1
Domain and range of the 6 trig functions
sin & cos
D: (-∞,∞)
R: [-1,1]
tan/cot
D: {x / x ≠ π/2 + κπ}
R: (-∞, ∞)
CAST Diagram
Q1: all positive
Q2: sin/csc positive
Q3: tan/cot positive
Q4: cos/sec positive
Periodicity summary
cos(θ + 2Kπ) = cosθ
sec(θ + 2Kπ) = secθ
sin(θ + 2Kπ) = sinθ
csc(θ + 2Kπ) = cscθ
tan(θ + Kπ) = tanθ
cot(θ + Kπ) = cotθ
Reciprocal identities
cotθ = 1/tanθ
secθ = 1/cosθ
cosecθ = 1/sinθ
Quotient identities
tanθ = sinθ/cosθ
cotθ = cosθ/sinθ
Pythagorean identites
sin²θ + cos²θ = 1
tan²θ + 1 = sec²θ
cot²θ + 1 = csc²θ
Even and Odd properties
sinθ is an odd function (symmetric with respect to origin):
-sinθ = sin(-θ)
cosθ is an even function (symmetric with respect to y-axis)
cos(-θ) = cosθ
Graphing trig functions
y = Asin(wx)
A: amplitude
Period (T): 2π/w
-sinx, reflect graph (s shape), also for sin(-x)
-cos(x), reflect graph (upside down), cos(-x) is simply cos(x)
always divide graph into 4 intervals when transforming
