y = a tan[b(x-c)] + d
Ex) y = 3 sin[2(x-5)] + 10
|a| = 3
d = 10
max = 13
min = 7
P = 360˚/2 = 180˚
y = a sin[b(x-c)] + d
amplitude = |a|
mean = d
max = d + |a|
min = d - |a|
P = 2π/b or 360˚/b
cos(θ-π/2) = sinθ
HT cosine π/2 right = sine graph
sin(θ+π/2) = cosθ
HT sine π/2 left = cosine graph
b = 2π/P or 360˚/P
Trig Graphs
Finding Angle
Determine Quadrants
for terminal arm
Find reference angle use special
triangles, unit circle, or calculator
Use domain restrictions to
find direction of rotation
Special Triangles
Find rotation angle
Finding Trig Ratios
Ex) 1. Given Point P(-3, 2) is on
the terminal arm of θ; Find cscθ
x^2 + y^2 = r^2
(-3)^2 + (2)^2 = 13
13 = r^2
r = √13
cscθ = √13/2
Primary & Reciprocal Trig Functions
Finding a reference angle (θr)
Ex) If the rotation angle is 135˚,
θr is 45˚ and also θr = π-θ
Ex) If the rotation angle is 70˚, the θr is
just the same as the rotation angle
Reference Angle
Ex) If the rotation angle is
300˚, θr = 60˚ so θr = 2π-θ
Ex) If the rotation angle is
230˚, θr = 50˚ so θr = θ-π
Trigonometry
Reference Angle θr is the angle
between the x-axis and terminal arm
Principal Angle(P) is the
smallest positive angle in
standard position
Coterminal Angle have
the same terminal arm
P˚+ 360n˚, n∈I or P+2πn, n∈I
Angle in Standard Position
Radian Measure
a = θ/r
θ = arc length/radius
Rad to Degree: Rad * (180˚/π)
Degree to Rad: Deg * (π/180˚)
π/2 = 1.57
Rotation counterclockwise: θ is positive
π = 3.14
3π/2 = 1.57
Rotation clockwise: θ is negative
2π = 6.28
Principal & Coterminal Angle
Coterminal Angle: Angles in standard position
that share the same terminal side. And there
are positive and negative coterminal angle.
Ex) If the θ = 45˚, the negative coterminal
angle = -375˚ and positive angle = 405˚
Positive coterminal
Negative coterminal angle
angle = θ + 360˚or θ + 2π
= θ - 360˚or θ - 2π
Ex) The principal angle (P) is
135˚ above the picture
Ex) a = 15 cm and radius is 4.3
cm and find for an angle
15/4.3 = 3.5 rad
3.5 rad * (180˚/π ) = 200.5˚
θ = 201˚