Trigonometry Lecture Notes
Section 1.1
Page 1 of 8
Section 1.1: Angles
Big Idea: Trigonometry is the study of triangles and how their angles and side lengths relate, and
the study of the so-called trigonometric functions of any angle. So, some basic ideas from
geometry about angles need to be reviewed.
Big Skill: You should be able to relate the measures of complementary and supplementary
angles, convert between DMS and DD representations for angle measure, and compute
coterminal angles.
Lines, Rays, and Segments
We can connect any two given two points A and B in a flat plane with a collection of colinear
points in three different ways:
 The points between A and B are called the line segment AB.

The points between A and B and then extending past either A or B (but not both) forever
are called either ray AB or ray BA. The point listed first is the endpoint of the ray.

The points between A and B and then extending past both A and B forever are called line
AB (or line BA).
Trigonometry Lecture Notes
Section 1.1
Page 2 of 8
Angles
 An angle consists of either two rays or two segments in a plane with a common endpoint.
 The common endpoint is called the vertex.
 The two rays or segments are called the sides of the angle.



The measure of an angle is a number associated with the amount of rotation about the
vertex required to rotate one side of the angle to the other side of the angle.
The initial side is the side from which the rotation begins.
The terminal side is the side at which the rotation ends.


A counterclockwise rotation has a positive measure.
A clockwise rotation has a negative measure.

Angles are named by stating a point on one side, the vertex point, and then a point on the
other side.
o Examples: angle ABC, or ABC, or CBA.

Angles are sometimes also named just by the vertex point.
Trigonometry Lecture Notes
Section 1.1
Page 3 of 8
Angle Measurement
 The most common unit for measuring angles is called the degree.
 The symbol for the degree is: 
 The Babylonians decided that there are 360 degrees (360) in one full rotation.
 A right angle is one-fourth of a full rotation and measures 90.
 A straight angle is one-half of a full rotation and measures 180.


An angle between 0 and 90 is called an acute angle.
An angle between 90 and 180 is called an obtuse angle.


Greek letters are often used to denote the measure of an angle:       
Strictly speaking, you should not mix up the name of an angle with its measure.
o If you are talking about an angle A (or A), you should say that the measure of
angle A is 85, or that m(angle A) = 85, or m(A) = 85.
o In reality, people usually just say or write A = 85.
Complementary and Supplementary Angles
 Complementary angles have measures that add up to 90.

Supplementary angles have measures that add up to 180.
Trigonometry Lecture Notes
Section 1.1
Page 4 of 8
Practice:
1. Find the complement and supplement of an angle measuring 38.
2. Find  in the picture below.
3. Find  in the picture below.
4. Find x in the picture below.
5. Find the measure of the smaller angle formed between the hands of a clock reading 2:38.
Trigonometry Lecture Notes
Section 1.1
Page 5 of 8
Fractions of a Degree
 An old-school technique for talking about angle measures less than a degree is to
subdivide the degree in the same way we subdivide time: into minutes and seconds.
1
 60  1
 One minute of angle is one-sixtieth of a degree: 1 
60
1
 60  1
 One second of angle is one-sixtieth of a minute: 1 
60
 Example of an angle measurement stated in degrees, minutes, and seconds (DMS):
322417
Converting from degrees, minutes, and seconds (DMS) to decimal degrees (DD)
 Example: convert 322417 to DD
 Divide the number of second by sixty to convert it to an equivalent number in minutes.
 1 
o 17 
  0.283 333
 60 
 Add that to the number of minutes, then divide by sixty again to convert to an equivalent
number in degrees.
 1 
o 24.283 333 
  0.404 722
 60 
 Add that number to the number of degrees, and round to an appropriate number of places:
o 322417  32.405
1
 0.000 278 . Thus, stating DMS measurements to the nearest
 Note that 1 
3600
thousandth of a degree won’t result in much round-off error, but stating your answer
beyond a ten-thousandth of a degree implies a precision not conveyed by mere seconds.
 Note: on a graphing calculator, you can type in a DMS measurement using the degree and
minute symbols found in the ANGLE menu, and the quotation marks for the seconds.
Practice:
6. Convert 992247 to DD.
7. 1574839 + 953642 =
Trigonometry Lecture Notes
Section 1.1
Page 6 of 8
8. Find the complement of an angle measuring 381218.
Converting from decimal degrees (DD) to degrees, minutes, and seconds (DMS)
 Example: convert 185.659 to DMS
 Multiply the decimal portion by sixty to convert it to an equivalent number in minutes.
 60 
o 0.6592 
  39.552
 1 
 Multiply the resulting decimal portion by sixty again to convert to an equivalent number
in seconds.
 60 
o 0.552 
  33.12
 1 
 Round to an appropriate number of places:
o 185.659  1853933
 Note: on a graphing calculator, you can type in a DD measurement and convert it to a
DMS measurement using the DMS function found in the ANGLE menu.
Practice:
9. Convert 22.128 to DMS.
Other Units of Angle Measure
 We will also learn about the radian unit for angle measurement, which is more widely
used in math and science fields.
 There is also a (now pretty much extinct) measurement unit called the “grad”, “gradian”,
or “gon”. There are 100 gradians in a right angle, and thus 400 gradians in a full rotation.
Old scientific calculators usually had a DRG button for converting between degrees,
radians, and gradians.
Trigonometry Lecture Notes
Section 1.1
Page 7 of 8
Standard Position
 An angle in standard position has its vertex at the origin of a Cartesian coordinate system
and its initial side on the positive x-axis.
 Thus, acute angles have their terminal sides in Quadrant I.
 Also, obtuse angles have their terminal sides in Quadrant II.
 A quadrantal angle has its terminal side on the x- or y-axis.
 Quadrantal angles have measure of 90, 180, 270, 360, 450, 540, 630, 720, …, 90, -180, -270, -360,-450, etc.
Coterminal Angles
 Coterminal angles have the same terminal side, but differ in how many rotations around
the circle were taken to get to that side.


The measures of coterminal angles differ by a multiple of 360.
Angles coterminal with a given angle measure of  will have measures given by
 + (360)n for any integer n.
Practice:
10. Find the coterminal angle of least possible positive measure for an angle of 1106.
Trigonometry Lecture Notes
Section 1.1
Page 8 of 8
11. Find the coterminal angle of least possible positive measure for an angle of -650.
12. If a record spins at 33 1/3 revolutions per minute, through how many degrees does it
rotate in 17 seconds?
13. If a gear rotates through 187 in 30 seconds, how many rotations does it make in an hour?