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in
rm
Te
id
al s
e
in
rm
Te
β
id
al s
e
α
Vertex
Initial Side
Counterclockwise rotation
Positive Angle
Vertex
Initial Side
Terminal side
Initial side
Terminal
side
Vertex
Vertex
90D angle;
Initial side
1
revolution
4
Theorem Arc Length
r
r
1 radian
if its vertex is at the origin of a rectangular
coordinate system and its initial side
coincides with with positive x - axis.
y
Terminal
side
θ
Clockwise rotation
Negative Angle
The angle formed by rotating the initial
side exactly once in the counterclockwise
direction until it coincides with itself (1
revolution) is said to measure 360 degrees,
D
abbreviated 360 .
An angle θ is said to be in standard position
For a circle of radius r, a central angle
of θ radians subtends an arc whose
length s is
s = rθ
Vertex
Initial side
x
Consider a circle of radius r.
Construct an angle whose vertex is at
the center of this circle, called the
central angle, and whose rays subtend
an arc on the circle whose length is r.
The measure of such an angle is 1
radian.
A triangle in which one angle is a right
angle is called a right triangle. The side
opposite the right angle is called the
hypotenuse, and the remaining two sides
are called the legs of the triangle.
c
b
90D
a
1
a
min
Ter
l si
de
c
θ
θ
a
Initial side
a = Adjacent = 5
b = Opposite = 12
Find the value of each of the six
trigonometric functions of the angle θ .
12
13
θ
Adjacent
b
c = Hypotenuse = 13
b = Opposite = 12
2
2
a +b = c
a 2 = 169 − 144 = 25
a =5
b 2 + a 2 = c2
c = Hypotenuse = 13
sinθ =
2
a 2 + 122 = 132
The six ratios of a right triangle are called
trigonometric functions of acute angles
and are defined as follows:
Function name Abbreviation
Value
sin θ
b/c
sine of θ
a/c
cosθ
cosine of θ
b/a
tan θ
tangent of θ
c/b
csc θ
cosecant of θ
c/a
secant of θ
sec θ
a /b
cotangent of θ
cot θ
Opposite
12 cscθ = Hypotenuse = 13
=
Opposite
12
Hypotenuse 13
Hypotenuse 13
Adjacent
5
secθ =
=
=
Adacent
5
Hypotenuse 13
Adjacent 5
Opposite 12
cot θ =
=
tanθ =
=
Opposite 12
Adjacent 5
cosθ =
c
b
b2 a 2 c2
+ =
c2 c2 c2
2
90D
a
2
 b +  a = 1
 c  c 
sin 2 θ + cos2 θ = 1
Theorem Complementary Angles
Theorem
Cofunctions of complementary angles
are equal.
2