Download Trigonometry Basics

Name: _________________
Trigonometry Basics
Date: ___________ Pd: ___
terminal side
Angle
vertex
Standard position
direction of rotation arrow
CCW is positive
CW is negative
initial side
vertex at the _______ and initial side along the ________
Positive and Negative
Angles
___________ angle
in standard position
___________ angle
in standard position
Angles “lie” in the quadrant in which the terminal side rests.
Examples: In what quadrant do the angles lie? Is the angle
positive or negative?
Quadrant
Adv Alg/Precal Trig Basics Notes
1
name given to an angle whose terminal side coincides with one
of the axes
-270
90
-360
0
0 -180
360
180
Quadrantal angle
270
-90
Quadrantal angle measures
for positive angles.
Quadrantal angle measures
for negative angles.
2 or more angles that have the same terminal side.
Finding coterminal angles – add/subtract 360 to the given
angle
Example: Find a positive and a negative angle coterminal with
the given angle.
Coterminal angles
a. 30
b. 170
Worksheet #1 – Trig Basics
Adv Alg/Precal Trig Basics Notes
2
c. 200
d. 400
e. 750
Many navigational systems use the degree-minute-second
(DMS) format.
DDDoMM SS
Example: 123o 4543
Degree-Minute-Second
(DMS)
Some calculations, however, use a decimal format for degree
measures. Decimal format is also easier to enter into
calculators.
Decimal  Degrees 
Minutes Seconds

60
3600
Example:
Converting from DMS Given: 123o 4543
to Decimal
Convert to Decimal: 123 
Converting from
Decimal to DMS
45
43

 123.762
60 3600
Degrees: the whole degrees to the left of the decimal point
Minutes: multiply the decimal degrees by 60; the minutes is
the whole number to the left of the decimal point
Seconds: multiply the decimal minutes by 60 and use the
product rounded to 1 decimal place
Example: Express 235.567 in DMS
Degrees: 235
Minutes: 0.567 * 60 = 34.02  34
Seconds: 0.02 * 60 = 1.2
Answer: 235.567 = 235 34 1.2
Worksheet #2 – Trig Basics
Adv Alg/Precal Trig Basics Notes
3
The fraction of a circle’s circumference that is intercepted by a
central angle. Arc length is measured in linear units (inches,
meters, cm, feet, etc.).
Arc Length
arc length

central angle
Another way to measure angles
Whenever an angle is used by itself (not inside a trig function),
the angle MUST be in radian units – NEVER degrees.
Definition: The central angle made by taking the radius of a
circle and wrapping it along the edge of the circle.
A
Length of arc AC = 2.75 cm
Radian
 = 1 radian
B
C
radius = 2.75 cm
So,
– if a central angle is 1.5 radians, then the intercepted arc is 1.5
times the radius of the circle
– if the length of an intercepted arc is 2.3 times the radius, then
the central angle is 2.3 radians.
π radians = 180
 degrees to radians, multiply by
Converting between
Radians and Degrees

180
of )
 from radians to degrees, multiply by
3 decimal places)
Adv Alg/Precal Trig Basics Notes
(usually leave in terms
4
180

(usually round to
Example: Convert 225 to radians
Answer:
   5
225 * 
radians

 180  4
Convert 3 radians to degrees
 180
Answer: 3 radians * 
 


  171.887

Worksheet #3 – Trig Basics
2 or more angles with the same
terminal side. In the diagram
angle  and angle  are
coterminal angles.


Very useful when evaluating
trigonometric functions (later).
Finding coterminal angles.
Coterminal Angles
 If angle is measured in degrees, add or subtract 360
 If angle is measured in radians, add or subtract 2
Examples:
Find a positive and a negative angle coterminal with the given
angle.
a. 75
Adv Alg/Precal Trig Basics Notes
b.
5
5
12
– the angle between the terminal side and the x-axis
– are always positive and less than 90
– useful when evaluating trig functions and solving trig
equations (later)
– found depending on quadrant. For positive angles,
 Quadrant I: reference angle = angle
 Quadrant II: reference angle = 180 – angle   angle 
 Quadrant III: reference angle = angle – 180
 angle   
 Quadrant IV: reference angle = 360 – angle
Reference Angles
 2 
angle 
Examples:
Find the reference angle for the following angles.
a. 175
b.
5
12
c. 300
d.
7
4
s  r
where s is the arc length, r is the radius, and 
is the central angle (IN RADIANS)
Example: A circle has a radius of 5 feet. Find the arc length
intercepted by a central angle measuring 2 radians.
Arc Length Theorem
s  r
Formula
s  (5 feet )(2radians )
Substitution
s  10 feet
Evaluation
Example: A central angle of 4 radians intercepts an arc with
length of 25 meters. What is the radius of the circle?
Adv Alg/Precal Trig Basics Notes
6
Sector Area
1
A  r 2
2
where A is the area of the sector, r is the
radius, and  is the central angle (IN
RADIANS)
Example: The minute hand of a clock is 4 inches long. How
much area will the minute hand sweep through in 20
minutes?
1
A  r 2
2
1
 2 
A  (4 in) 2 

2
 3 
16 2
A
in
3
A  16.76 in 2
Worksheet #4 – Trig Basics
Adv Alg/Precal Trig Basics Notes
7
2 radians

Formula
Find θ:

60 minutes 20 minutes
2
Substitution

radians
3
Evaluation
I feel a need… a need for speed!
Linear speed 
change in distance
time
Angular speed 
change in central angle ( )
time
Rotational speed 
change in revolutions
time
Conversions Conversions between speed units
between
speed units ① 1 revolution = 1 circumference (distance travelled)
(rotational)
(linear)
②
③
1 circumference = 2  radians
(linear)
(angular)
2 radians = 1 revolution
(angular)
(rotational)
Hint on performing unit conversions:
Multiply by
the units you want
the units you have
Example: Convert 2.1 miles to feet
2.1 miles *
5280 feet (units you want )
 11,088 feet
1 mile (units you have)
Example: Convert 35 miles to inches
35 miles *
Adv Alg/Precal Trig Basics Notes
5280 feet (units you want ) 12 inches (units you want )
*
 2,217,600 inches
1 mile (units you have)
1 foot (units you have)
8
Example:
A bicycle with wheels that have a radius of 20 inches is travelling at 15
miles per hour. How many revolutions per second are the wheels turning?
Analysis: You are given a linear speed. The given linear units are miles
so you need to convert to inches (units of the radius). Time is
given in hours, but the final answer requires it to be in seconds
so you need to convert hours to seconds. Finally, once the
linear units and time units are set, convert from linear units to
rotational units using relationship ① above.
Solution:
15 miles
1 hour
Original
units
1 hour
60 min
1 min
60 sec
Convert hours
to seconds
5280 ft 12 in
1 mile 1 ft
1 revolution
2π(20 in)
= 1.03 rev/sec
Convert linear
Convert linear speed to
units to same
rotational speed
units as radius
Example:
The radius of a CD is 2.25 in. The CD player rotates the CD at 400 rpm
(revolutions per minute). What is the linear speed (in feet per second) of a
speck of dust on the outer edge of the CD?
Worksheet #5 – Trig Basics
Adv Alg/Precal Trig Basics Notes
9