### angle

```Chapter 4 Vocabulary
Section 4.1 vocabulary
An angle is
determined by a
rotating ray
its endpoint.
The starting point
of the ray is the
initial side of
the angle.
The position of the
ray after the
rotation is the
terminal side of
the angle.
The endpoint of
the ray is the
vertex of the
angle.
When an angle fits a
coordinate system in which
the origin is the vertex of
the angle, and the initial
side coincides with the
positive x-axis that angle is
in standard position.
Positive angles
are generated by
counterclockwise
rotation.
Negative angles
are generated by
a clockwise
rotation.
Two angles that have
the same initial and
terminal sides are
called coterminal
angles.
The measure of an
angle is determined
by the amount of
rotation from the
initial side to the
terminal side.
A central angle is
an angle whose
vertex is the
center of the
circle.
of a central angle Ѳ that
intercepts an arc s equal in
length to the radius r of the
circle.
Ѳ = s / r , where Ѳ is
Angles between
0 and ∏ / 2 are
called acute
angles.
Angles between
∏/2 and ∏ are
called obtuse
angles.
A way to measure angles
is in degrees where 1
degree is equivalent to a
rotation of 1/360 of a
complete revolution
Complementary
90 degrees
or∏/2.
Supplementary
angles sum to
equal 180
degrees or ∏.
Linear speed
Linear speed = Arc length / time
Angular speed
Angular speed = central angle/ time
The unit circle
Given by the equation :
2
2
X +y =1
Definitions of Trigonometric functions
•
•
•
•
•
•
Sin (t) = y
Cos(t) = x
Tan(t) = y/x
Csc(t) = 1/y
Sec(t) = 1/x
Cot(t) = x/y
• A function f is periodic if there
exists a positive real number c
such that :
f(t + c) = f(t)
For all t in the domain of f.
The least number c for which f is
periodic is called the period of
f.
Even/ odd trig functions
Even
• cos(-t) = cos(t)
• sec(-t) = sec(t)
odd
• Sin(-t) = -sin(t)
• tan(-t) = -tan(t)
• csc(-t) = -csc(t)
• cot(-t) = -cot(t)
Section 4.3 Vocabulary
Right triangle def. of Trig Functions
•
•
•
•
•
•
Sin(Ѳ) = opp/hyp
Csc(Ѳ) = hyp/opp
Sines of special angles
•Sin(30) =sin(∏/6) = ½
•Sin (45) = sin(∏/4) =
√2/2
•Sin(60) = sin(∏/3) = √3/2
Cosines of special angles
• Cos(30) = cos(∏/6) = √3/2
• Cos(45) = cos(∏/4) = √2/2
• Cos(60) = cos(∏/3) = ½
Tangents of Special angles
• Tan(30) = tan(∏/6) = √3/3
• Tan(45) = tan(∏/4) = 1
• Tan(60) = tan(∏/3) = √3
Reciprocal Identities
•
•
•
•
•
•
Sin(Ѳ) = 1/csc(Ѳ)
Cos(Ѳ) = 1/ sec(Ѳ)
Tan(Ѳ) = 1/cot(Ѳ)
Csc(Ѳ) = 1/sin(Ѳ)
Sec(Ѳ) = 1/cos(Ѳ)
Cot(Ѳ) = 1/tan(Ѳ)
Quotient identities
• Tan(Ѳ) = sin(Ѳ) / cos(Ѳ)
• Cot(Ѳ) = cos(Ѳ) / sin(Ѳ)
Pythagorean Identities
2
•Sin (Ѳ)
2
cos (Ѳ)
+
=1
2
2
•1 + tan (Ѳ) = sec (Ѳ)
2
2
•1 + cot (Ѳ) = csc (Ѳ)
Angle of elevation
•The angle from
the horizontal
up to the object
Angle of Depression
•The angle from
the horizontal
downward to the
object.
Section 4.4 Vocabulary
Definitions of Trig
Functions
Sin Ѳ = y/r
cos Ѳ = x/r
Tan Ѳ = y/x
Cot Ѳ = x/y
Sec Ѳ = r/x
Csc Ѳ = r/y
Reference Angle
• Let Ѳ be an angle in standard
position. Its reference angle is
the acute angle Ѳ’ formed by
the terminal side of V and the
horizontal axis.
Section 4.6 Vocabulary
Amplitude
• The amplitude of y = a sin(x)
And y = a cos(x)
Represents half of the distance
between the max and the min values
of the function, and is given by
Amplitude = |a|
Period
• The b be a positive real
number. The period of y = a
sin(bx) and t = a cos(bx) is
given by
Period = 2∏/b
Damping factor
•In the function f(x) = x
sin(x), the factor x is
called the damping
factor.
Section 4.7 Vocabulary4
Inverse sine function
y = sin (x) has a unique inverse
function called inverse sine
function. It is denoted by
Y =arcsin(x) or y =
-1
sin
(x)
Inverse cosine function
y = cos (x) has a unique inverse
function called inverse cosine
function. It is denoted by
Y =arccos(x) or y =
-1
cos
(x)
Inverse tangent function
y = tan (x) has a unique inverse
function called inverse
tangent function. It is denoted
by
Y =arctan(x) or y = tan-1 (x)
```