Pre Calculus

1
Pre Calculus

• Day 37

2
Plan for the Day

• Section 4.1 Intro to Trigonometry
• Trigonometry - Measurement of Triangles
• Key Terms
• Radians, Degrees and Conversions
• The Calculator
• Co-terminal Angles
• Arc Measures
• Short Quizzes Every Day

3
What You Should Learn

• Describe angles.
• Use radian measure.
• Use degree measure.
• Convert between radians and degrees
• Use angles to model and solve real-life
  problems.

4
Angles ?, ?, ? or A, B, C

• Angle Rotation of a ray about its endpoint
• Initial Side The starting position of the ray
• Terminal Side The position of the ray after
  rotation
• Standard Position An angle placed on the
  coordinate plane with the initial side coincides
  with the positive x-axis
• Positive angles generated by counterclockwise
  rotation
• Negative angles generated by clockwise rotation
• Co terminal angles with the same initial and
  terminal sides
• Measure of an Angle The amount of rotation from
  the initial side to the terminal side
• Central Angle An angle whose vertex is the
  center of a circle

5
Angles

• As derived from the Greek language, the word
  trigonometry means measurement of triangles.
• Initially, trigonometry dealt with relationships
  among the sides and angles of triangles.
• An angle is determined by rotating a ray
  (half-line) about its endpoint.

6
Angles
An angle is formed by two rays that have a common
endpoint called the vertex. One ray is called the
initial side and the other the terminal side. The
arrow near the vertex shows the direction and the
amount of rotation from the initial side to the
terminal side.
7
Angles

• The endpoint of the ray is the vertex of the
  angle.
• This perception of an angle fits a coordinate
  system in which the origin is the vertex and the
  initial side coincides with the positive x-axis.
  
• Such an angle is in standard position.

Angle in standard position
8
Degree Measure

• One way to measure angles is in terms of degrees,
  denoted by the symbol ?.
• A measure of one degree (1?) is equivalent to a
  rotation of of a complete revolution about
  the vertex.
• To measure angles, it is convenient to mark
  degrees on the circumference of a circle.

Figure 4.13
9
Measuring Angles Using Degrees
The figures below show angles classified by
their degree measurement. An acute angle measures
less than 90º. A right angle, one quarter of a
complete rotation, measures 90º and can be
identified by a small square at the vertex. An
obtuse angle measures more than 90º but less than
180º. A straight angle has measure 180º.
10
Angles of the Rectangular Coordinate System

• An angle is in standard position if
• its vertex is at the origin of a rectangular
  coordinate system and
• its initial side lies along the positive x-axis.

11
When an angle is in standard position, it is
important to know what quadrant your angle has
its terminal side

• What are the quadrants??

12
Circle with Degrees
90o
Quadrant I 0olt?lt 90o
Quadrant II 90olt?lt 180o
180o
0o
360o
Quadrant IV 270olt?lt 360o
Quadrant III 180olt?lt 270o
270o
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Angle Measures - Degrees

• Positive Angles
• 0olt?lt 90o Quadrant I (acute)
• ? 90o Y-axis (right)
• 90olt?lt180o Quadrant II (obtuse)
• ? 180o X-axis (straight)
• 180olt?lt270o Quadrant III
• ? 270o Y-axis
• 270olt?lt360o Quadrant IV
• ? 360o X-axis

• Negative Angles
• 0ogt?gt -90o Quadrant IV
• ? -90o Y-axis
• -90ogt?gt-180o Quadrant III
• ? -180o X-axis
• -180ogt?gt-270o Quadrant II
• ? -270o Y-axis
• -270ogt?gt-360o Quadrant I
• ? -360o X-axis

14
What quadrant is it in?

• a) 30o b) 120o c) -45o
• d) 190o e) 370o f) -95o
• g) 280o h) 820o i) -745o

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Coterminal Angles
Angle A
Angle B
Angle A and B are coterminal
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Coterminal Angles

• An angle of xº is coterminal with angles of
• xº k 360º where k is an integer.

17
Co-terminal

• ? 360o ? 360o
• Name two co terminal angles, one positive and one
  negative
• 30o
• 30o 360o 390o 30o - 360o -330o
• 120o
• 120o 360o 480o 120o - 360o -240o
• -45o
• -45o 360o 315o -45o - 360o -405o

18
Degrees, Minutes, Seconds

• Measurement in Degrees
• A minute is 1/60 of a degree denoted with
• A second is 1/60 of a minute denoted with
• Ex 150o 35 25
• Where is it on the calculator??
• Converting from decimal form to DMS

19
Radian Measure

• A second way to measure angles is in radians.
  This type of measure is especially useful in
  calculus.
• To define a radian, you can use a central angle
  of a circle, one whose vertex is the center of
  the circle.

Arc length radius when ? 1 radian
Figure 4.5
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Radian - Definition

• One radian is the angle subtended at the center
  of a circle by an arc of circumference that is
  equal in length to the radius of the circle.
• In terms of a circle it can be seen as the ratio
  of the length of the arc subtended by two radii
  to the radius of the circle.
• http//en.wikipedia.org

21
How many Radians are in a circle?

• Because the circumference of a circle is 2? r
  units, it follows that a central angle of one
  full revolution (counterclockwise) corresponds to
  an arc length of
  s 2? r.
• If a circle has a radius of 1 what is its
  circumference?
• Circumference pd or 2pr
• Circumference 2p(1) or 2p
• There are 2p radians in every circle
• How about a half circle? A quarter circle?

22
Radian Measure

• Because the units of measure for s and r are the
  same, the ratio s / r has no unitsit is simply a
  real number.
• Moreover, because 2? ? 6.28, there are just over
  six radius lengths in a full circle.

Figure 4.6
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Radian Measure

• Because the radian measure of an angle of one
  full revolution is 2?, you can obtain the
  following.

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Radian Measure

• These and other common angles we will frequently
  use.

25
Circle with Degrees and Radians
90o or p/2
180o or p
0o or 0
360o or 2p
270o or 3p/2
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Angle Measures - Radians
One radian is the measure of a central angle ?
that intercepts an arc s equal in length of the
radius r of the circle.

• Positive Angles
• 0lt?ltp/2 Quadrant I (acute)
• ?p/2 Y-axis
• (right)
• p/2lt?lt? Quadrant II
• (obtuse)
• ? ? X-axis
• (straight)
• ?lt?lt3p/2 Quadrant III
• ?3p/2 Y-axis
• 3p/2lt?lt2? Quadrant IV
• ?2? X-axis

• Negative Angles
• 0gt?gt-p/2 Quadrant IV
• ?-p/2 Y-axis
• -p/2gt?gt-? Quadrant III
• ? -? X-axis
• -?gt?gt-3p/2 Quadrant II
• ?-3p/2 Y-axis
• -3p/2gt?gt-2? Quadrant I
• ?-2? X-axis

27
What quadrant is it in?

• a) p/3 b) p/12 c) -p/3
• d) 5p/6 e) 3p/4 f) -4p/3
• g) p/6 h) -5p/6 i) 7p/3

28
Radian Measure

• Two angles are coterminal if they have the same
  initial and terminal sides. For instance, the
  angles 0 and 2? are coterminal, as are the angles
  ? / 6 and 13? / 6.
• You can find an angle that is coterminal to a
  given angle ? by adding or subtracting 2? (one
  revolution).
• A given angle ? has infinitely many coterminal
  angles. For instance, ? ? / 6 is coterminal
  where n is an integer.

29
Coterminal Angles

• An angle of x is coterminal with angles of
• x k 2p
• where k is an integer.

30
Co-terminal

• ? 2p
• ? 2p
• Name two coterminal angles, one positive and one
  negative
• a) p/2 b) p/12 c) -p/4

31
Calculator Issues

• Mode you must know what you are working with!!!
• Xo means degrees
• X with no symbol is radians it is the default!
• Can be written Xr

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Other Angle Properties

• Two positive angles ? and ? are complementary
  (complements of each other) if their sum is ? / 2
  (radians) or 90o (degrees).
• Two positive angles are supplementary
  (supplements of each other) if their sum is ?
  (radians) or 180o (degrees).

Supplementary angles
Complementary angles
33
Finding Complements and Supplements

• For an xº angle, the complement is a 90º xº
  or ? / 2 xº angle.
• Thus, the complements measure is found by
  subtracting the angles measure from 90º or ? /
  2.
• For an xº angle, the supplement is a 180º xº
  or ? xº angle.
• Thus, the supplements measure is found by
  subtracting the angles measure from 180º or ? .

34
Find the Complement and Supplement

1. p/3
2. p/12
3. 55o
4. 5p/6
5. 75o

35
Degree Measure

• When no units of angle measure are specified,
  radian measure is implied.
• For instance, if you write ? 2, you imply that
  ? 2 radians.
• Most radian measures you work with will have p in
  them.

36
Conversion between Degrees and Radians

• Using the basic relationship ? radians 180º,
• To convert degrees to radians, multiply degrees
  by (? radians) / 180?
• To convert radians to degrees, multiply radians
  by 180? / (? radians)

37
Convert to the other form

1. p/3
2. p/12
3. 55o
4. -5p/6
5. -75o

38
Length of a Circular Arc

• Let r be the radius of a circle and ? the non-
• negative radian measure of a central angle
• of the circle. The length of the arc
• intercepted by the central angle is
• s r ?

39
Finding Arc Length

• A circle has a radius of 4 inches. Find the
  length of the arc intercepted by a central angle
  of 240?.

40
Solution

• First, we convertto radians
• Then, using a radius of r 4 inches, you can
  find the arc length to bes r?

41
Try these

• Find the length of the arc of a circle with a
  radius of 5 and a central angle of 5 radians.
• Find the length of the arc of a circle with a
  radius of 3 and a central angle of p/4 radians.
• Find the length of the radius of a circle with an
  arc length of 10 and a central angle of p/6
  radians.
• Find the length of the arc of a circle with a
  radius of 7 and a central angle of 50o.

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Homework 19

• Section 4.1Page 269 -271 7, 9, 17, 20, 21, 23,
  27, 31, 38, 39, 43, 47, 56, 57, 63, 66, 71, 77,
  81, 85, 89, 91