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Transcript
1.1 -- Angles
Wednesday, August 07, 2013
12:53 PM
Basic Terminology:
Line AB
Segment AB
Ray AB
Angles in Standard Position
Animation of terminal side of an angle
Sketch a picture of the following angle measures.
1. 240⁰
2.
50⁰
Naming an Angle:
• It can be named by its vertex or
• by using 3 points, making sure the vertex
is the middle letter.
4.
3. 150⁰
B•
A
Degree Measure: We assign 360 degrees to a full rotation of a ray.
Chapter 1 Page 1
•
C
300⁰
The Greek letter Ө (theta) is used to name each angle.
Other Greek letters such as (alpha) and β (beta) are also used often.
Acute angle
Right Angle
Obtuse Angle
Complementary Angles
Straight Angle
Supplementary Angles
Ex 1: Finding the complement and the Supplement of an Angle
Classroom Example 1:
For an angle measuring 40˚
For an angle measuring 55˚
a) Find the complement
a) Find the complement
b) Find the supplement
b) Find the supplement
Ex2:
Find the measure of each angle in the give
figures.
2)
1)
4x˚
6x˚
6x˚
3x˚
3)
4)
2x˚
7x˚
3x˚
Chapter 1 Page 2
2x˚
The measure of an angle can be written
where it is read "the measure of angle A = 35"
Traditionally, portions o a degree have been measured with minutes and seconds. One
minute is written 1'. One second is written 1".
or
or
12˚ 42' 38" = 12 degrees, 42 minutes 38 seconds
Ex3: Calculating with Degrees, Minutes, and Seconds
Perform each calculation:
1) 51˚ 29' + 32˚ 46'
2) 90˚
Classroom Ex3: Perform each calculation.
1) 28˚ 35' + 63˚ 52'
2) 180
73˚ 12'
117˚ 29'
Because calculators are so prevalent, angles are commonly measured in decimal degrees. We will be
converting them to degrees, minutes, and seconds (D˚ M' S"). For example 12.4238˚ (your calculator
may be able to make those conversions for you.
Ex4: Converting between Decimal Degrees and D˚ M' S"
1) Convert 74˚ 08' 14"
2) convert 34.817˚
to decimal degrees to the nearest thousandth.
to D˚ M' S" to the nearest second
Chapter 1 Page 3
Classroom Ex4:
1) Convert 105˚ 20' 32"
2) convert 85.263˚
to decimal degrees to the nearest thousandth.
to D˚ M' S" to the nearest second
Quad Angle Animation
Quadrant Angles
's in standard position whose terminal sides lie on the x-axis or the y-axis such as angles
with measures 90˚, 180˚, and 270˚ are Quadrant Angles.
Coterminal Angles
Coterminal Angle Animation
Angles that continue past 360˚ are coterminal angles. Angles with measures of 60˚ and 420˚
have the same initial side and terminal side, but they differ in the amount of rotation. Their
measures differ by multiples of 360˚.
Ex5: Finding Measures of Coterminal Angles
Find the angles of least positive measure that are coterminal with each angle.
1) 908˚
2) 75˚
3) 800˚
1) 1106˚
5)
150˚
6)
603˚
Ex6: Analyzing the Revolutions of a CD player
CD players always spin at the same speed. Suppose a CD player makes 480 revolutions per
min. Through how many degrees will a point on the edge of a CD move in 2 sec?
Chapter 1 Page 4
Classroom Ex6:
A wheel made 270 revolutions per min. Through how many degrees will a point on the edge
of the wheel move in 5 seconds?
Chapter 1 Page 5
1.2 -- Angle Relationships and Similar Triangles
Wednesday, August 07, 2013
12:55 PM
Geometric Properties
• Vertical Angles are congruent
Parallel Lines form angles that are congruent or supplementary
1) The corr
are
2) The alt int
3) The alt ext
are
are
4) The cons int
are supp
Ex1: Finding Angle Measures
Find the measures of angles 1, 2, 3, and 4 given that the lines are parallel.
1
(3x + 2)˚ 1 2
2
3
4
(9x + 9)˚
3 (7x - 5)˚
(5x + 40)˚
4
Triangles
• The angle sum of a triangle is 180
Ex2: Applying the Angle Sum Triangle Property
1. Find the measure of the 3rd angle of a triangle that has 2 angles with measures of 48˚ and 61˚.
2. Find the measure of the 3rd angle of a triangle that has 2 angles with measures of 33˚ and 26˚.
3. One angle is 36⁰ 43' and the 2nd angle is 58⁰ 29', find the measure of the 3rd angle.
Chapter 1 Page 6
ANGLES
SIDES
Types of triangles.
Similar Triangles:
Triangles that are similar have exactly the same shape but not necessarily the same size.
Similar
Similar
Similar and
Congruent
Conditions for Similar Triangles
1) All the corresponding angles must be congruent
2) All of the corresponding sides must be proportional (that is, the ratios of the corresp sides must be equal)
Ex3: Finding angle measures in similar triangles
Find the measure of all angles, given that
ΔABC is similar to ΔDEF
104˚
45˚
Chapter 1 Page 7
Ex4: Finding Side Lengths in Similar Δs
24
16
The triangles are similar, find
the length of each side.
8
32
12
y
12
8
10
x
The exterior angle of a triangle is equal to the sum of the 2 remote interior angles of the triangle.
1
2
4
3
Chapter 1 Page 8
Because if you add angle 4
to each side the total will
be 180⁰. Therefore they
are equal
Quiz Review?
Thursday, August 08, 2013
2:40 PM
Chapter 1 Page 9
1.3 -- Trig Functions
Wednesday, August 07, 2013
12:56 PM
Trigonometric Functions
Ex1: Finding Function Values of an Angle
The terminal side of an angle θ in standard position passes through the point (8, 15). Find
the 6 trig functions of angle θ.
sin θ =
csc θ =
cos θ =
sec θ =
tan θ =
cot θ =
Classroom Ex1:
The terminal side of an angle θ in standard position passes through the point (12, 5). Find
the 6 trig functions of angle θ.
Ex2: Finding Function Values of an Angle
Chapter 1 Page 10
Ex2: Finding Function Values of an Angle
The terminal side of an angle θ in standard position passes through the point ( 3, 4). Find
the values of the 6 trig functions of angle θ.
x= 3
y= 4
r=
θ
-3
-4
(-3, -4)
Classroom Ex2:The terminal side of an angle θ in standard position passes through the point
(8, 6). Find the values of the 6 trig functions of angle θ.
** We can also find the trig function values of an angle if we know the equation of the line coinciding with the
terminal ray. Recall from algebra that the graph of the equation
is the line that passes through the origin. (in standard form)
Ex3: Finding Function Values of an Angle
Find the 6 trig functions of the angle θ in standard position, if the terminal side of θ is defined by x + 2y = 0, and x
≥ 0. (1) Pick any value of x (the easiest one is the coefficient of y - because then you won't get fraction answers)
then (2) plug it into the equation and solve for y. (3) Use that (x, y) point to solve for r.
Classroom Ex3:Find the 6 trig functions of the angle θ in standard position, if the terminal
Chapter 1 Page 11
Classroom Ex3:Find the 6 trig functions of the angle θ in standard position, if the terminal
side of θ is defined by 3x 2y = 0, and x ≤ 0.
Slope-intercept form of a line: y = mx + b
In Ex3, the equation x + 2y = 0 can be written as,
So the slope is
. Notice that
In general,
it is true that
The reciprocal of 0 is undefined.
Quadrantal Angles
Remember that quadrantal angles 0˚, 90˚, 180˚, 270˚, and 360˚. When determining trig function values of
quadrantal angles, the figure below can help find the ratios. Because any point on the terminal side can be
used, it is convenient to choose the point 1 unit from the origin, with r = 1. (in ch3 this idea will extend to the
unit circle.)
(0, 1)
x = -1
y=0
r=1
(-1, 0)
x=0
y=1
r=1
90˚
x=0
y = -1
r=1
(1, 0)
x=1
y=0
r=1
sin 90 =
csc 90 =
cos 90 =
sec 90 =
tan 90 =
cot 90 =
(0, -1)
Ex4: Find the values of the 6 trig functions for an angle θ in standard position with terminal
side through ( 3,0)
Chapter 1 Page 12
Use the trig function values of quadrantal angles to evaluate each expression.
If you see cot2 90⁰ it means (cot 90⁰)2
Chapter 1 Page 13
1.4 -- Using the Definitions of the Trig Functions
Wednesday, August 07, 2013
12:56 PM
Identities are equations that are true for all values of the variables for which all expressions are defined.
Reciprocal identities:
For all angles θ for which both functions are defined, the following identities hold.
CAUTION Be sure not to use the inverse trig keys on your calculator to find reciprocals. For ex:
Ex1: Using the Reciprocal Identities
1) Find cos θ, given that sec θ =
2) Find sin θ, given that csc θ =
3) Find tan θ, given that cot θ = 4
4) Find sec θ, given that cos θ =
Sign and Ranges of Function Values
All Students Take Calculus
Q1 -- All functions are positive
QII -- Sin and csc are positive
QIII -- Tan and cot are positive
QIV -- Cos and sec are positive
Ex2: Determining Signs of Functions of Nonquadrant Angles
Chapter 1 Page 14
Ex2: Determining Signs of Functions of Nonquadrant Angles
Determine the quadrant of the given angle and name which trig functions are positive in that
quadrant.
1. 87˚
2) 300˚
3)
200˚
1. 54˚
5) 260˚
6)
60˚
Ex3: Identifying the Quadrant of an Angle
Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions.
1)
2)
4)
3)
-1 ≤ sin θ ≤ 1
-1 ≤ cos θ ≤ 1
sec θ ≤ -1 OR sec θ ≥ 1
csc θ ≤ -1 OR csc θ ≥ 1
tan θ can be any real #
cot θ can be any real #
Ex4: Deciding Whether a Value is in the Range of a Trig Function
Decide whether each statement is possible or impossible
1) sin θ = 2.5
2) cot θ = 0.999
Chapter 1 Page 15
2) tan θ = 110.47
3) sec θ = 0.6
5) cos θ = 1.7
6) csc θ = 0
2) cot θ = 0.999
5) cos θ = 1.7
6) csc θ = 0
Ex5: Finding All Function Values Given One Value and the Quadrant
1) Suppose that angle θ is in Q II and sin θ = . Find the values of the other 5 trig functions.
Classroom Ex5: Suppose that angle θ is in Q III and tan θ = . Find the values of the other 5
trig functions.
Pythagorean Identities
From x2 + y2 = r2 we can derive 3 new identities.
As before, we are only given one form of each identity. However, with algebra we can
rewrite each of these. For example.
is equivalent to
** it is important to be able to transform these identities quickly and to recognize their equivalent forms!
Chapter 1 Page 16
Quotient Identities:
So let's look at the quotient of
** note: the denominators cannot = 0
Ex6: Using Identities to Find Function Values
Find sin θ and tan θ, given that
and sin θ > 0
Classroom Ex6: Find cos θ and tan θ, given that sin θ =
and cos θ > 0.
Ex7: Using Identities to Find Function Values
** we can use x2 + y2 = r2
Find sin θ and cos θ, given that tan θ = and θ is in QIII.
Using Pythagorean and quotient identities OR
x 2 + y 2 = r2
Classroom Ex7: Find sin θ and cos θ, given that cot θ =
Chapter 1 Page 17
and θ is in QII.
Classroom Ex7: Find sin θ and cos θ, given that cot θ =
Chapter 1 Page 18
and θ is in QII.
Test Review?
Thursday, August 08, 2013
2:40 PM
Chapter 1 Page 19