The document provides an overview of angles and their measurements in geometry and trigonometry, including definitions, types of angles, and methods for measuring them. It discusses standard position angles, quadrantal angles, coterminal angles, reference angles, and the units used for angle measurement (revolutions, degrees, and radians). Additionally, it introduces the concept of the unit circle, its equation, and examples of points in relation to the circle.

1. The Unit Circle
Grade 11 - STEM

2. Angles and Angle Measurement
•In geometry, an angle is defined as the
union of two non-collinear rays with a
common endpoint. The definition
restricts the angle measures between 0
and 180 exclusive.

3. Angles and Angle Measurement
•In trigonometry, angles will be discussed
in a different perspective. Angles will be
thought of as any amount of rotation
generated when a ray is rotated about
its endpoint, hence, the necessity to
label the sides of an angle.

4. Angles and Angle Measurement
• The initial position of the ray will be referred to as
initial side of the angle, and the position of the ray at
the end of its rotation will be called terminal side of the
angle. The arrowhead is pointed at the terminal side.
The endpoint of the ray is the vertex of the angle.

5. Angles and Angle Measurement
•The direction of rotation of a ray also affects angle
measure. If the rotation of the ray is counterclockwise,
a positive angle is formed. If the rotation of the ray is
clockwise, a negative angle is formed.

6. Example:
Draw the following angle measure
1. 45 4. 0°
2. 150° 5. -30°
3. 400° 6. -500°

7. Angles in Standard Position
•An angle is said to be in standard position if its vertex
is at the point of origin of a Cartesian Coordinate
Plane, and its initial side is along the positive side of
the x-axis.

8. Quadrantal Angles
•A quadrantal angle is an angle in standard
position whose terminal side lies on the x-
axis and y-axis.

9. Coterminal Angles
•A coterminal angles that are both in
standard position and have the same
terminal sides.

10. Example:
Determine the measure of the smallest positive angle
coterminal with the given angle.
1. 42 3. -213°
2. 108° 4. 635°

11. Reference Angles
•The reference angle for a given angle is the
positive acute angle formed by the terminal
side of the given angle and the x-axis.

12. Example:
Find the reference angles of the following:
1. 42 3. 113°
2. 221° 4. -410°

13. Units of Angle Measure
There are three units of angle measure that
are commonly used:
I. Revolution
II. Degree Measure
III. Radian Measure

14. Units of Angle Measure
A revolution (rev) is formed
when the initial side of an angle
rotates completely around its vertex.
Consequently, the angle’s terminal
side and initial side coincide.

15. Units of Angle Measure
The Babylonians developed the degree measure at
around 2000 B.C. It was based on a sexagesimal
numeration system. To one complete rotation of a ray, or
to a revolution, 360° is assigned. One degree is equal to
60 minutes, and 1 minute is equal to 60 seconds.
1 rev = 360°
1° = 60’
1’ = 60”

16. Example:
Change each angle measure in decimals to minutes
and seconds.
1. 22.4 2. 14.21°

17. Units of Angle Measure
One radian (rad) is the measure of a central angle
subtended by an arc of a circle that is equal to the
radius of the circle.

18. Converting Angle Measure
•To convert degree measure to radian measure,
multiply the number of degrees by , then simplify.
•To convert radian measure to degree measure,
multiply the number of radians by , then simplify.

19. Example:
Convert each radian measure to degree measure.
1. 2.

20. Example:
Convert each degree measure to radian measure.
1. 45° 2. -108°

21. The Unit Circle
•A unit circle is a circle whose center is at the origin
and with a radius of 1 unit. Every point on the unit
circle satisfies the equation . Some of points on the
unit circle are ( 1, 0 ), ( 0, 1 ), (-1, 0), and (0, -1)

22. Example:
Determine whether each point is inside, on, or
outside the unit circle.
1. 3.
2. 4. (1.1, -0.1)

23. Thank you
for
listening