“TRIGONOMETRY”
Derived from Greek language
Means “measurement of triangles”
Initially, trig dealt with relationships among the
sides and angles of triangles and was used in
astronomy, navigation, and surveying
In 17th century, a different perspective arose:
trig relationships were viewed as functions of
real numbers with applications including
rotations and vibrations
ANGLE
Determined by rotating a ray about its endpoint
Can be labeled with uppercase letters (A, B, or C) or
with Greek letters (a, b, or q)
ANGLE
When viewed with respect
to the coordinate system,
an angle is in standard
position if
 Its vertex is at the origin
 Its initial side lies along the
positive x-axis
ANGLE
POSITIVE ANGLES
N E G AT I V E A N G L E S
Generated by
counterclockwise
rotation
Generated by
clockwise rotation
ANGLE
LIES IN A QUADRANT
Q UA D R A N TA L A N G L E
Terminal side of angle
in standard position
lies in a quadrant
Terminal side of angle
lies ON the x-axis or
y-axis
Examples
Examples
MEASURE OF ANGLES
AMOUNT OF ROTATION FROM THE INITIAL SIDE TO
THE TERMINAL SIDE
DEGREES,
°
Full rotation
(revolution) = 360°
1
1° =
of a rotation
360
about the vertex
RADIANS
Useful in calculus
Measure of the central
angle of a circle that
intercepts an arc
equal in length to the
radius of the circle
(draw picture)
Full revolution is 2p≈6.28
RELATIONSHIP (CONVERSION) BETWEEN DEGREES
AND RADIANS
DEGREES TO
RADIANS
Multiply stated degrees
by
𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
180°
RADIANS TO
DEGREES
Multiply stated radians
180°
by
𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
Examples
Examples
ANGLES ARE CLASSIFIED BY DEGREES OR RADIANS
ACUTE ANGLE
RIGHT ANGLE
Measures between 0°
and 90° or
between 0 radians
𝜋
and radians
Measures exactly 90°
𝜋
or
2
0° < q < 90°
0<q<
𝜋
2
2
q=
𝜋
2
ANGLES ARE CLASSIFIED BY DEGREES OR RADIANS
OBTUSE ANGLE
STRAIGHT ANGLE
(LINE)
Measures between 90°
and 180° or between
𝜋
radians and p
2
radians
Measures exactly 180°
or p
90° < q < 180°
q= p
𝜋
2
<q< p
q = 180°
COTERMINAL ANGLES
Two or more angles with the same initial and terminal
side, but possibly different rotations
Every angle has infinitely many coterminal angles.
𝜃 ± 360°𝑘, 𝑤ℎ𝑒𝑟𝑒 𝑘 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
𝜃 ± 2𝜋𝑘, 𝑤ℎ𝑒𝑟𝑒 𝑘 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
LENGTH OF A CIRCULAR ARC
Let r be the radius of a circle and q the nonnegative
radian measure of a central angle of the circle. The
length of the arc intercepted by the central angle is
s = rq
Draw picture
Examples
ASSIGNMENT
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