Lesson 7-1
Objective: To learn the
foundations of trigonometry.
Trigonometry
The branch of mathematics that
studies right triangles. It deals with
the relationship between the sides
and the angles.
In order to do this you must also
understand the relationship between
angles and circles.
Angles
• An angle is formed by two rays with
a common endpoint. (geometry
definition)
• An angle is the result of a rotation
of a ray about its endpoint.
(trigonometry definition)
Standard Position
When the initial side is on the positive x
axis and the endpoint is on the origin
Then the angle is in standard position.
Positive Angle – Standard
Form
α
Negative Angle – Standard
Form
β
Quadrant I
• Where the terminal side lays is
where the angle is said to lie.
Between 0o and
90o or 0 and 
2
Quadrant II
Between 90o and
180o or  and
2

Quadrant III
Between 180o and
3
o
270 or and 2

Quadrant IV
Between 270o and
3
o
360 or 2 and 2
Quadrantal
• When the terminal side lies on an
axis it is called a quadrantal.
Angular Measurement
1
360
• Degree of a complete rotation
in the counterclockwise direction.
1o
Angular Measurement
1
360
• Degree of a complete rotation
in the counterclockwise direction.
10o
Angular Measurement
1
360
• Degree of a complete rotation
in the counterclockwise direction.
45o
Angular Measurement
1
360
• Degree of a complete rotation
in the counterclockwise direction.
90o
Angular Measurement
1
360
• Degree of a complete rotation
in the counterclockwise direction.
150o
Angular Measurement
1
360
• Degree of a complete rotation
in the counterclockwise direction.
225o
Angular Measurement
1
360
• Degree of a complete rotation
in the counterclockwise direction.
315o
Angular Measurement
1
360
• Degree of a complete rotation
in the counterclockwise direction.
359o
Review -Classifying Angles
Acute Angle - measure
greater than 0 degrees and
less than 90 degrees
Review-Classifying Angles
Obtuse - measure more than 90
degrees and less than 180
degrees (in Quadrant II)
Review - Classifying Angles
Right - measure 90 degrees
(Quadrantal)
Radian Measure
• To talk about trigonometric
functions, it is helpful to move to a
different system of angle measure,
called radian measure.
• A radian is the measure of a
central angle whose intercepted
arc is equal in length to the radius
of the circle.
23
©Carolyn C. Wheater, 2000
Radian Measure
• There are 2 radians in a full rotation once around the circle ( is half a
rotation)
• There are 360° in a full rotation (180 °
in half a rotation)
• To convert from degrees to radians or
radians to degrees, use the proportion:
degrees = radians

180°
24
©Carolyn C. Wheater, 2000
Sample Problems
• Find the degree
measure
3
equivalent of
4
radians.
degrees radians


360
2
d
3 4


360
2
2d  270
d  135

 Find the radian
measure
equivalent of
210°
degrees radians


360
2
210
r


360
2
360r  420
420 7
r

360
6
Radians (cont’d)
• Typically the angle is referred to as
θ (theta)
• Some standard angle conversions

•
radians = 180o
• 1 radian =(180 )o

• 1o =

180
radians
Examples
• Convert from degree to radian
measure:
• -210o
• 390o
Examples
• Convert from radian measure to
degrees.
•
•
9 radians
2
4
3 radians