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Trigonometric Ratios and
Functions
1
TRIGONOMETRY
• primarily a branch of mathematics that deals with triangles, mostly right
triangles. In particular the ratios and relationships between the triangle's sides
and angles. It has two main ways of being used:
• In Geometry: In its geometry application, it is mainlyused to solve triangles,
usually right triangles. That is, given some angles and side lengths, we can find
some or all the others.
• Analytically: In a more advanced use, the trigonometric ratios such as Sine and
Tangent, are used as functions in equations and are manipulatedusing algebra.
TRIGONOMETRY
Trigonometry is over three thousand years old. The ancient Egyptians,
Babylonians, and Greeks developed trigonometry to find lengths of the
sides of triangles and the measures of their angles. In Egypt trigonometry
was used to establish land boundaries after the annual flood of the Nile
River. In Babylonia, it was used in astronomy.
ANGLES AND THEIR MEASURE
• A ray is part of the line that is
described as a ‘half- line’ and that
in which one of the one has an end
point on the other side and is goes
on infinitely. The point from which
the ray originates is called the
initial point of the ray.
When two rays share a common
initial point, they form an angle and
the common initial point is calledthe
vertex of the angle. Two examples of
what are commonly thought of as
angles are:
MEASURE OF AN ANGLE
The measure of an angle is a number
which indicates the amount of
rotation that separates the rays of
the angle
Greek Letters are being used to
measure theangles. Some
examples are below:
𝑎 𝖰 𝜽
alpha beta theta
DEGREE MEASURE
Degree Measure – commonly used system to measure
angles. Quantities measured in degrees are denoted by the
symbol “°”. One completerevolution as shown below is
360°. Thus, half of a revolution (a straight angle) measures
180°. A quarter of a revolution (a right angle) measures
90°. “90°”
“360°” “180°”
CONVERTING
DEGREES TO DMS
• In this system, one degree is divided
equally into sixty minutes, and in turn,
each minute is divided equally into sixty
seconds.
• In Illustration:
• 1° = 60’ (read as 60 minutes)
• 1′ = 60” (read as 60 seconds)
• 1° = 3600”
“90°”
“360°” “180°”
CONVERTING DEGREES
TO DMS
1. Convert a measure of 42.125 to DMS System
42.125° = 42° + 0.125°
42.125° = 42° + 0.125°
60°
1′
42.125° = 42° + 7.5′
42.125° = 42° + 7′
+ 0.5°
60°
1′
42.125° = 42° + 7′
+ 30′′
42.125° = 42°7′
30′′
CONVERTING DEGREES
TO DMS
1. Convert a 117°15′
45′′to degree measure
117°15′45′′ = 117° + 15′ + 45′′
117°15′45′′ = 117° + 15
′
1°
60′
+ 45′′
1°
3600°
117°15′45" = 117° + 0.25° + 0.0125
117°15′45" = 117.2626°
CONVERTING DEGREES
TO DMS
3. Convert a 78°8′
14′′to degree measure
78°8′14′′ = 117° + 15′ + 45′′
78°8′14′′= 78° + 8
′
1°
60′
+ 14′′
1°
3600°
78°8′14′′= 78° + 0.13333° + 0.0039°
78°8′
14′′ ≈ 74.137°
CONVERTING DEGREES
TO DMS
4. Convert a 34°. 817 DMS
34.817° = 34° + .817°
34.817° = 34°+.817(60′)
34.817° = 34° + 49.02
34.817° = 34° + 49′ + .02 60′′
34.817° = 34°49′
1.2′′
10.2 Angles and Radian
Measure
After this lesson…
• I can draw angles in standard position.
• I can explain the meaning of radian measure.
• I can convert between degrees and radians.
• I can use radian measure to find arc lengths and the area of a sector.
12
10.2 Angles and Radian Measure
• Angles in Standard Position
• Vertex on origin
• Initial Side on positive x-axis
• Measured counterclockwise
10.2 Angles and Radian Measure
• Coterminal Angles
• Different angles (measures) that
have the same terminal side
• Found by adding or subtracting
multiples of 360°
15
Radian and Degree Measure
Positive and negative angles
Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–4
Section 4.1, Figure 4.3, Positive and
Negative Angles, pg. 248
When sketching angles,
always use an arrow to
show direction.
16
Find one positive angle and one negative angle that are coterminal with (a)
−45° and (b) 395°.
There are many such angles, depending on what multiple of 360° is added or subtracted.
17
10.2 Angles and Radian Measure
• Draw an angle with the given
measure in standard position. Then
find one positive coterminal angle
and one negative coterminal angle.
• 65°
• 300°
• −900°
−125°
10.2 Angles and Radian Measure
• Radian measure
• Another unit to measure angles
• 1 radian is the angle when the arc
length = the radius
• There are 2π radians in a circle
10.2 Angles and Radian Measure
• To convert between degrees and
radians use fact that
• 180° = π
• Special angles
10.2 Angles and Radian Measure
• Convert the degree measure to radians, or the radian measure to degrees.
• 135°
• -50°
•
5𝜋
4
•
40°
10.2 Angles and Radian Measure
• Sector
• Slice of a circle
• Arc Length
• 𝑠 = 𝑟θ
• 𝜃 must be in radians!
• Area of Sector
• 𝐴 =
1
2
𝑟2𝜃
• 𝜃 must be in radians!
10.2 Angles and Radian Measure
• Find the length of the outfield fence
if it is 220 ft from home plate.
• Find the area of the baseball field.
• Try 535#29 Find the Drone Search
Area
10.1 Right
Triangle
Trigonometry
After this lesson…
• I can define the six
trigonometric
functions.
• I can evaluate
trigonometric
functions.
• I can use
trigonometric
functions to find
side lengths of right
triangles.
24
10.1 Right Triangle Trigonometry
• If you have a right triangle, there are
six ratios of sides that are always
constant
• sin 𝜃 =
opposite
hypotenuse
• cos 𝜃 =
adjacent
hypotenuse
• tan 𝜃 =
opposite
adjacent
• csc 𝜃 =
hypotenuse
opposite
• sec 𝜃 =
hypotenuse
adjacent
• cot 𝜃 =
adjacent
opposite
SOH
CAH
TOA
10.1 Right Triangle Trigonometry
• Evaluate the six trigonometric functions of the angle 𝜃.
10.1 Right Triangle Trigonometry
• In a right triangle, 𝜃 is an acute angle
and cos 𝜃 =
7
10
. What is sin 𝜃?
• Try 526#9
If sin 𝜃 =
7
11
, what are other trig
functions?
10.1 Right Triangle Trigonometry
• Special Right Triangles
• 30° - 60° - 90°
• 45° - 45° - 90°
10.1 Right Triangle Trigonometry
• Use the diagram to solve the right triangle if…
• B = 60°, a = 7
• A = 32°, b = 10
• Try 526#33
A = 43°, b = 31
10.1 Right Triangle Trigonometry
• Find the distance between Powell
Point and Widforss Point.
• Try 527#37 Find the width of the
river.
10.3 Trigonometric Functions
of Any Angle
After this lesson…
• I can evaluate trigonometric functions given a point on an angle.
• I can evaluate trigonometric functions using the unit circle.
• I can find and use reference angles to evaluate trigonometric functions.
• I can solve real-life problems involving projectiles.
31
10.3 Trigonometric Functions
of Any Angle
• Think of a point on the terminal side
of an angle
• You can draw a right triangle with
the x-axis
• sin 𝜃 =
𝑦
𝑟
csc 𝜃 =
𝑟
𝑦
• cos 𝜃 =
𝑥
𝑟
sec 𝜃 =
𝑟
𝑥
• tan 𝜃 =
𝑦
𝑥
cot 𝜃 =
𝑥
𝑦
• Unit Circle
• r = 1
10.3 Trigonometric Functions
of Any Angle
• Evaluate the six trigonometric functions of θ.
Try 542#3
10.3 Trigonometric Functions
of Any Angle
• Quadrantal Angles
• Evaluate the six trigonometric functions of θ.
• θ = 180°
10.3 Trigonometric Functions
of Any Angle
• Reference Angle
• Angle between terminal side and
x-axis
• Has the same values for trig
functions as 1st quadrant angles
• You just have to add the negative
signs
All
Sin
Cos
Tan
10.3 Trigonometric Functions
of Any Angle
• Sketch the angle. Then find its
reference angle.
• 150°
• Try 542#15
23𝜋
4
10.3 Trigonometric Functions
of Any Angle
• Evaluate cos(-60°) without a
calculator
• Try 542#25
sin(−150°)
10.3 Trigonometric Functions
of Any Angle
• Estimate the horizontal distance traveled by a Red Kangaroo who jumps at an
angle of 8° and with an initial speed of 53 feet per second (35 mph).
10.4 Graphing Sine and
Cosine Functions
After this lesson…
• I can identify characteristics of sine and cosine functions.
• I can graph transformations of sine and cosine functions.
39
10.4 Graphing Sine and Cosine Functions
• Work with a partner.
• a. Complete the table for y = sin x, where x is an angle measure in radians.
• b. Plot the points (x, y) from part (a). Draw a smooth curve through the points
to sketch the graph of y = sin x. Make several observations about the graph.
40
10.4 Graphing Sine and Cosine Functions
• 𝑦 = sin 𝑥
• Starts at 0
• Amplitude = 1
• Period = 2π
• 𝑦 = cos 𝑥
• Starts at 1
• Amplitude = 1
• Period = 2π
10.4 Graphing Sine and Cosine Functions
• Transformations
• 𝑦 = 𝑎 sin 𝑏 𝑥 − ℎ + 𝑘
• |a| = amplitude = vertical stretch
• If a < 0, then reflect over x-axis
• b = horizontal shrink
• Period 𝑇 =
2𝜋
𝑏
• h = horizontal shift
• k = vertical shift
• Midline 𝑦 = 𝑘
10.4 Graphing Sine and Cosine Functions
• Graphing sine and cosine
1. Identify the amplitude, period, horizontal shift, and vertical shift
2. Draw the midline, y = k
3. Find the 5 key points (3 zeros, 1 max, 1 min)
4. Draw the graph
43
• Identify the amplitude and period. • Try 551#7
44
10.4 Graphing Sine and Cosine Functions
• Graph 𝑓 𝑥 = 2 sin 𝑥
10.4 Graphing Sine and Cosine Functions
• Try 551#13
Graph 𝑦 = sin 2𝜋𝑥
10.4 Graphing Sine and Cosine Functions
• Graph 𝑦 = 2 sin 𝑥 −
𝜋
2
10.4 Graphing Sine and Cosine Functions
• Try 552#39
Graph
𝑔 𝑥 = −4 cos 𝑥 +
𝜋
4
− 1
10.5 Graphing Other
Trigonometric Functions
After this lesson…
• I can identify characteristics of tangent, cotangent, secant, and cosecant
functions.
• I can graph tangent and cotangent functions.
• I can graph secant and cosecant functions.
49
10.5 Graphing Other Trigonometric Functions
• Work with a partner.
• a. Complete the table for y = tan x, where x is an angle measure in radians.
• b. Plot the points (x, y) from part (a). Then sketch
the graph of y = tan x. Make several observations
about the graph.
50
10.5 Graphing Other Trigonometric Functions
• 𝑦 = tan 𝑥
• Period = π
• 𝑇 =
𝜋
𝑏
• Asymptotes where tangent
undefined,
𝜋
2
,
3𝜋
2
10.5 Graphing Other Trigonometric Functions
• 𝑦 = cot 𝑥
• Period = π
• 𝑇 =
𝜋
𝑏
• Asymptotes at 0, π, 2π
10.5 Graphing Other Trigonometric Functions
• Graph 𝑦 = tan
𝑥
4
10.5 Graphing Other Trigonometric Functions
• Try 560#7
Graph 𝑔 𝑥 =
1
2
tan 𝜋𝑥
10.5 Graphing Other Trigonometric Functions
• 𝑦 = csc 𝑥
• Period = 2π
• 𝑇 =
2𝜋
𝑏
• Asymptotes where sine = 0
• 0, 𝜋, 2𝜋
10.5 Graphing Other Trigonometric Functions
• 𝑦 = sec 𝑥
• Period = 2π
• 𝑇 =
2𝜋
𝑏
• Asymptotes where cosine = 0
•
𝜋
2
,
3𝜋
2
10.5 Graphing Other Trigonometric Functions
• Graph 𝑦 = 2 csc 𝑥 +
𝜋
2
10.5 Graphing Other Trigonometric Functions
• Try 560#15
Graph 𝑔 𝑥 = csc 4𝑥
10.6 Modeling with
Trigonometric Functions
After this lesson…
• I can write and graph trigonometric functions using frequency.
• I can write trigonometric functions for a given graph.
• I can find a trigonometric model for a set of data using technology.
59
10.6 Modeling with Trigonometric Functions
• Trigonometric functions are periodic
• Useful for modeling oscillating motions or repeated patterns
• Period (T)
• Time of 1 cycles
• Unit: s/cycle = s
• Frequency (f)
• Cycles per 1 second
• Unit: cycles/s = Hz
60
10.6 Modeling with Trigonometric Functions
• Find the frequency
𝑦 = 2 cos 3𝑥
• Try 568#3
𝑦 = sin 3𝜋𝑥
61
10.6 Modeling with Trigonometric Functions
• Write Trigonometric Models
• Find the amplitude
• Find the period
• If the situation starts at zero, use sine
• If starts increasing +
• If starts decreasing −
• If the situation starts at a maximum or minimum use cosine
• If starts at max +
• If starts at min −
62
10.6 Modeling with Trigonometric Functions
• An audiometer produces a pure tone with a frequency f of 1000 hertz (cycles
per second). The maximum pressure P produced by the tone is 20
millipascals. Write a sine model that gives the pressure P as a function of the
time t (in seconds).
63
10.6 Modeling with Trigonometric Functions
• Try 568#9
Frequency is measured in hertz, or cycles per second. The lowest frequency of
sounds that can be heard by humans is 20 hertz. The maximum pressure P
produced from a sound with a frequency of 20 hertz is 0.02 millipascal. Write
a sine model that gives the pressure P as a function of the time t (in seconds).
64
10.6 Modeling with Trigonometric Functions
• Write a function for the sinusoid
shown.
65
10.6 Modeling with Trigonometric Functions
• Try 569#11
Write a function for the sinusoid
shown.
66
10.6 Modeling with Trigonometric Functions
• Two people swing jump ropes. The highest point of the middle of each rope is
80 inches above the ground and the lowest point is 2 inches above the
ground. Each rope makes 2 revolutions per second. Write a model for the
height h (in inches) of one of the ropes as a function of the time t (in seconds)
given that the rope is at its lowest point when t = 0.
67
10.6 Modeling with Trigonometric Functions
• The tables show the average
monthly low temperatures D (in
degrees Fahrenheit) in Erie,
Pennsylvania, where t = 1 represents
January. Write a model that gives D
as a function of t and interpret the
period of its graph. Use technology.
• Press STAT, Edit… enter points
• Press STAT  CALC, SinReg
68
10.6 Modeling with Trigonometric Functions
• Try 569#19
The tables show the average
monthly low temperatures D (in
degrees Fahrenheit) in Las Vegas,
Nevada, where t = 1 represents
January. Write a model that gives D
as a function of t and interpret the
period of its graph. Use technology.
69
10.7 Using Trigonometric
Identities
After this lesson…
• I can evaluate trigonometric functions using trigonometric identities.
• I can simplify trigonometric expressions using trigonometric identities.
• I can verify trigonometric identities.
70
10.7 Using Trigonometric Identities
• On unit circle
• 𝑥 = cos 𝜃
• 𝑦 = sin 𝜃
• 𝑥2 + 𝑦2 = 1
• cos2 𝜃 + sin2 𝜃 = 1
• Trigonometric Identity
• Statement showing relationship
between two quantities that are
always =
71
10.7 Using Trigonometric Identities
• Reciprocal Identities
• sin 𝑢 =
1
csc 𝑢
csc 𝑢 =
1
sin 𝑢
• cos 𝑢 =
1
sec 𝑢
sec 𝑢 =
1
cos 𝑢
• tan 𝑢 =
1
cot 𝑢
cot 𝑢 =
1
tan 𝑢
• Quotient Identities
• tan 𝑢 =
sin 𝑢
cos 𝑢
cot 𝑢 =
cos 𝑢
sin 𝑢
• Pythagorean Identities
• sin2
𝑢 + cos2
𝑢 = 1
• tan2 𝑢 + 1 = sec2 𝑢
• 1 + cot2 𝑢 = csc2 𝑢
10.7 Using Trigonometric Identities
• Even/Odd Identities
• cos −𝑢 = cos 𝑢
• sec −𝑢 = sec 𝑢
• sin −𝑢 = − sin 𝑢
• tan −𝑢 = − tan 𝑢
• csc −𝑢 = − csc 𝑢
• cot −𝑢 = − cot 𝑢
• Cofunction Identities
• sin
𝜋
2
− 𝑢 = cos 𝑢
• cos
𝜋
2
− 𝑢 = sin 𝑢
• tan
𝜋
2
− 𝑢 = cot 𝑢
• cot
𝜋
2
− 𝑢 = tan 𝑢
• sec
𝜋
2
− 𝑢 = csc 𝑢
• csc
𝜋
2
− 𝑢 = sec 𝑢
10.7 Using Trigonometric Identities
• Given that sin 𝜃 = −
5
13
and 𝜋 < 𝜃 <
3𝜋
2
, find the values of the other five
trigonometric functions of θ.
• Try 575#3
Given that tan 𝜃 = −
3
7
and
𝜋
2
< 𝜃 <
𝜋, find the values of the other five
trigonometric functions of θ.
74
10.7 Using Trigonometric Identities
• Simplify (1 + cos θ )(1 − cos θ ) • Try 575#9
sin 𝑥 cot 𝑥
75
10.7 Using Trigonometric Identities
• Verify Trigonometric Identities
• Show that trig identities are true by turning one side into the other side
• Guidelines
1. Work with 1 side at a time. Start with the more complicated side.
2. Try factor, add fractions, square a binomial, etc.
3. Use fundamental identities
4. If the above doesn’t work, try rewriting in sines and cosines
5. Try something!
10.7 Using Trigonometric Identities
• Verify
1
sec 𝜃−tan 𝜃
= sec 𝜃 + tan 𝜃 • Try 575#21
cos
𝜋
2
− 𝑥 cot 𝑥 = cos 𝑥
77
10.8 Using Sum and
Difference Formulas
After this lesson…
• I can evaluate trigonometric expressions using sum and difference formulas.
• I can simplify trigonometric expressions using sum and difference formulas.
• I can solve trigonometric equations using sum and difference formulas.
78
10.8 Using Sum and Difference Formulas
• Sum and Difference Formulas
• sin 𝑢 + 𝑣 = sin 𝑢 cos 𝑣 + cos 𝑢 sin 𝑣
• sin 𝑢 − 𝑣 = sin 𝑢 cos 𝑣 − cos 𝑢 sin 𝑣
• cos 𝑢 + 𝑣 = cos 𝑢 cos 𝑣 − sin 𝑢 sin 𝑣
• cos 𝑢 − 𝑣 = cos 𝑢 cos 𝑣 + sin 𝑢 sin 𝑣
• tan 𝑢 + 𝑣 =
tan 𝑢+tan 𝑣
1−tan 𝑢 tan 𝑣
• tan 𝑢 − 𝑣 =
tan 𝑢−tan 𝑣
1+tan 𝑢 tan 𝑣
10.8 Using Sum and Difference Formulas
• Find the exact value of sin 75°. • Try 581#1
tan −15°
80
10.8 Using Sum and Difference Formulas
• Find cos(a − b) given that
cos 𝑎 = −
12
13
with
𝜋
2
< 𝑎 < 𝜋 and
sin 𝑏 =
3
5
with 0 < 𝑏 <
𝜋
2
.
• Try 581#9
Find sin 𝑎 + 𝑏 given that cos 𝑎 =
4
5
with 0 < 𝑎 <
𝜋
2
and sin 𝑏 = −
15
17
with
3𝜋
2
< 𝑏 < 2𝜋.
81
10.8 Using Sum and Difference Formulas
• Simplify the expression cos(x − π). • Try 581#17
cos 𝑥 + 2𝜋
82
10.8 Using Sum and Difference Formulas
• Solve cos 𝑥 +
𝜋
4
+ cos 𝑥 −
𝜋
4
= 1
for 0 ≤ 𝑥 < 𝜋.
• Try 581#23 Solve on 0 ≤ 𝑥 < 2𝜋
2 sin 𝑥 − 1 = 0
83

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Trigonometric Ratios and Functions 2.pptx

  • 2. TRIGONOMETRY • primarily a branch of mathematics that deals with triangles, mostly right triangles. In particular the ratios and relationships between the triangle's sides and angles. It has two main ways of being used: • In Geometry: In its geometry application, it is mainlyused to solve triangles, usually right triangles. That is, given some angles and side lengths, we can find some or all the others. • Analytically: In a more advanced use, the trigonometric ratios such as Sine and Tangent, are used as functions in equations and are manipulatedusing algebra.
  • 3. TRIGONOMETRY Trigonometry is over three thousand years old. The ancient Egyptians, Babylonians, and Greeks developed trigonometry to find lengths of the sides of triangles and the measures of their angles. In Egypt trigonometry was used to establish land boundaries after the annual flood of the Nile River. In Babylonia, it was used in astronomy.
  • 4. ANGLES AND THEIR MEASURE • A ray is part of the line that is described as a ‘half- line’ and that in which one of the one has an end point on the other side and is goes on infinitely. The point from which the ray originates is called the initial point of the ray. When two rays share a common initial point, they form an angle and the common initial point is calledthe vertex of the angle. Two examples of what are commonly thought of as angles are:
  • 5. MEASURE OF AN ANGLE The measure of an angle is a number which indicates the amount of rotation that separates the rays of the angle Greek Letters are being used to measure theangles. Some examples are below: 𝑎 𝖰 𝜽 alpha beta theta
  • 6. DEGREE MEASURE Degree Measure – commonly used system to measure angles. Quantities measured in degrees are denoted by the symbol “°”. One completerevolution as shown below is 360°. Thus, half of a revolution (a straight angle) measures 180°. A quarter of a revolution (a right angle) measures 90°. “90°” “360°” “180°”
  • 7. CONVERTING DEGREES TO DMS • In this system, one degree is divided equally into sixty minutes, and in turn, each minute is divided equally into sixty seconds. • In Illustration: • 1° = 60’ (read as 60 minutes) • 1′ = 60” (read as 60 seconds) • 1° = 3600” “90°” “360°” “180°”
  • 8. CONVERTING DEGREES TO DMS 1. Convert a measure of 42.125 to DMS System 42.125° = 42° + 0.125° 42.125° = 42° + 0.125° 60° 1′ 42.125° = 42° + 7.5′ 42.125° = 42° + 7′ + 0.5° 60° 1′ 42.125° = 42° + 7′ + 30′′ 42.125° = 42°7′ 30′′
  • 9. CONVERTING DEGREES TO DMS 1. Convert a 117°15′ 45′′to degree measure 117°15′45′′ = 117° + 15′ + 45′′ 117°15′45′′ = 117° + 15 ′ 1° 60′ + 45′′ 1° 3600° 117°15′45" = 117° + 0.25° + 0.0125 117°15′45" = 117.2626°
  • 10. CONVERTING DEGREES TO DMS 3. Convert a 78°8′ 14′′to degree measure 78°8′14′′ = 117° + 15′ + 45′′ 78°8′14′′= 78° + 8 ′ 1° 60′ + 14′′ 1° 3600° 78°8′14′′= 78° + 0.13333° + 0.0039° 78°8′ 14′′ ≈ 74.137°
  • 11. CONVERTING DEGREES TO DMS 4. Convert a 34°. 817 DMS 34.817° = 34° + .817° 34.817° = 34°+.817(60′) 34.817° = 34° + 49.02 34.817° = 34° + 49′ + .02 60′′ 34.817° = 34°49′ 1.2′′
  • 12. 10.2 Angles and Radian Measure After this lesson… • I can draw angles in standard position. • I can explain the meaning of radian measure. • I can convert between degrees and radians. • I can use radian measure to find arc lengths and the area of a sector. 12
  • 13. 10.2 Angles and Radian Measure • Angles in Standard Position • Vertex on origin • Initial Side on positive x-axis • Measured counterclockwise
  • 14. 10.2 Angles and Radian Measure • Coterminal Angles • Different angles (measures) that have the same terminal side • Found by adding or subtracting multiples of 360°
  • 15. 15 Radian and Degree Measure Positive and negative angles Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–4 Section 4.1, Figure 4.3, Positive and Negative Angles, pg. 248 When sketching angles, always use an arrow to show direction.
  • 16. 16
  • 17. Find one positive angle and one negative angle that are coterminal with (a) −45° and (b) 395°. There are many such angles, depending on what multiple of 360° is added or subtracted. 17
  • 18. 10.2 Angles and Radian Measure • Draw an angle with the given measure in standard position. Then find one positive coterminal angle and one negative coterminal angle. • 65° • 300° • −900° −125°
  • 19. 10.2 Angles and Radian Measure • Radian measure • Another unit to measure angles • 1 radian is the angle when the arc length = the radius • There are 2π radians in a circle
  • 20. 10.2 Angles and Radian Measure • To convert between degrees and radians use fact that • 180° = π • Special angles
  • 21. 10.2 Angles and Radian Measure • Convert the degree measure to radians, or the radian measure to degrees. • 135° • -50° • 5𝜋 4 • 40°
  • 22. 10.2 Angles and Radian Measure • Sector • Slice of a circle • Arc Length • 𝑠 = 𝑟θ • 𝜃 must be in radians! • Area of Sector • 𝐴 = 1 2 𝑟2𝜃 • 𝜃 must be in radians!
  • 23. 10.2 Angles and Radian Measure • Find the length of the outfield fence if it is 220 ft from home plate. • Find the area of the baseball field. • Try 535#29 Find the Drone Search Area
  • 24. 10.1 Right Triangle Trigonometry After this lesson… • I can define the six trigonometric functions. • I can evaluate trigonometric functions. • I can use trigonometric functions to find side lengths of right triangles. 24
  • 25. 10.1 Right Triangle Trigonometry • If you have a right triangle, there are six ratios of sides that are always constant • sin 𝜃 = opposite hypotenuse • cos 𝜃 = adjacent hypotenuse • tan 𝜃 = opposite adjacent • csc 𝜃 = hypotenuse opposite • sec 𝜃 = hypotenuse adjacent • cot 𝜃 = adjacent opposite SOH CAH TOA
  • 26. 10.1 Right Triangle Trigonometry • Evaluate the six trigonometric functions of the angle 𝜃.
  • 27. 10.1 Right Triangle Trigonometry • In a right triangle, 𝜃 is an acute angle and cos 𝜃 = 7 10 . What is sin 𝜃? • Try 526#9 If sin 𝜃 = 7 11 , what are other trig functions?
  • 28. 10.1 Right Triangle Trigonometry • Special Right Triangles • 30° - 60° - 90° • 45° - 45° - 90°
  • 29. 10.1 Right Triangle Trigonometry • Use the diagram to solve the right triangle if… • B = 60°, a = 7 • A = 32°, b = 10 • Try 526#33 A = 43°, b = 31
  • 30. 10.1 Right Triangle Trigonometry • Find the distance between Powell Point and Widforss Point. • Try 527#37 Find the width of the river.
  • 31. 10.3 Trigonometric Functions of Any Angle After this lesson… • I can evaluate trigonometric functions given a point on an angle. • I can evaluate trigonometric functions using the unit circle. • I can find and use reference angles to evaluate trigonometric functions. • I can solve real-life problems involving projectiles. 31
  • 32. 10.3 Trigonometric Functions of Any Angle • Think of a point on the terminal side of an angle • You can draw a right triangle with the x-axis • sin 𝜃 = 𝑦 𝑟 csc 𝜃 = 𝑟 𝑦 • cos 𝜃 = 𝑥 𝑟 sec 𝜃 = 𝑟 𝑥 • tan 𝜃 = 𝑦 𝑥 cot 𝜃 = 𝑥 𝑦 • Unit Circle • r = 1
  • 33. 10.3 Trigonometric Functions of Any Angle • Evaluate the six trigonometric functions of θ. Try 542#3
  • 34. 10.3 Trigonometric Functions of Any Angle • Quadrantal Angles • Evaluate the six trigonometric functions of θ. • θ = 180°
  • 35. 10.3 Trigonometric Functions of Any Angle • Reference Angle • Angle between terminal side and x-axis • Has the same values for trig functions as 1st quadrant angles • You just have to add the negative signs All Sin Cos Tan
  • 36. 10.3 Trigonometric Functions of Any Angle • Sketch the angle. Then find its reference angle. • 150° • Try 542#15 23𝜋 4
  • 37. 10.3 Trigonometric Functions of Any Angle • Evaluate cos(-60°) without a calculator • Try 542#25 sin(−150°)
  • 38. 10.3 Trigonometric Functions of Any Angle • Estimate the horizontal distance traveled by a Red Kangaroo who jumps at an angle of 8° and with an initial speed of 53 feet per second (35 mph).
  • 39. 10.4 Graphing Sine and Cosine Functions After this lesson… • I can identify characteristics of sine and cosine functions. • I can graph transformations of sine and cosine functions. 39
  • 40. 10.4 Graphing Sine and Cosine Functions • Work with a partner. • a. Complete the table for y = sin x, where x is an angle measure in radians. • b. Plot the points (x, y) from part (a). Draw a smooth curve through the points to sketch the graph of y = sin x. Make several observations about the graph. 40
  • 41. 10.4 Graphing Sine and Cosine Functions • 𝑦 = sin 𝑥 • Starts at 0 • Amplitude = 1 • Period = 2π • 𝑦 = cos 𝑥 • Starts at 1 • Amplitude = 1 • Period = 2π
  • 42. 10.4 Graphing Sine and Cosine Functions • Transformations • 𝑦 = 𝑎 sin 𝑏 𝑥 − ℎ + 𝑘 • |a| = amplitude = vertical stretch • If a < 0, then reflect over x-axis • b = horizontal shrink • Period 𝑇 = 2𝜋 𝑏 • h = horizontal shift • k = vertical shift • Midline 𝑦 = 𝑘
  • 43. 10.4 Graphing Sine and Cosine Functions • Graphing sine and cosine 1. Identify the amplitude, period, horizontal shift, and vertical shift 2. Draw the midline, y = k 3. Find the 5 key points (3 zeros, 1 max, 1 min) 4. Draw the graph 43
  • 44. • Identify the amplitude and period. • Try 551#7 44
  • 45. 10.4 Graphing Sine and Cosine Functions • Graph 𝑓 𝑥 = 2 sin 𝑥
  • 46. 10.4 Graphing Sine and Cosine Functions • Try 551#13 Graph 𝑦 = sin 2𝜋𝑥
  • 47. 10.4 Graphing Sine and Cosine Functions • Graph 𝑦 = 2 sin 𝑥 − 𝜋 2
  • 48. 10.4 Graphing Sine and Cosine Functions • Try 552#39 Graph 𝑔 𝑥 = −4 cos 𝑥 + 𝜋 4 − 1
  • 49. 10.5 Graphing Other Trigonometric Functions After this lesson… • I can identify characteristics of tangent, cotangent, secant, and cosecant functions. • I can graph tangent and cotangent functions. • I can graph secant and cosecant functions. 49
  • 50. 10.5 Graphing Other Trigonometric Functions • Work with a partner. • a. Complete the table for y = tan x, where x is an angle measure in radians. • b. Plot the points (x, y) from part (a). Then sketch the graph of y = tan x. Make several observations about the graph. 50
  • 51. 10.5 Graphing Other Trigonometric Functions • 𝑦 = tan 𝑥 • Period = π • 𝑇 = 𝜋 𝑏 • Asymptotes where tangent undefined, 𝜋 2 , 3𝜋 2
  • 52. 10.5 Graphing Other Trigonometric Functions • 𝑦 = cot 𝑥 • Period = π • 𝑇 = 𝜋 𝑏 • Asymptotes at 0, π, 2π
  • 53. 10.5 Graphing Other Trigonometric Functions • Graph 𝑦 = tan 𝑥 4
  • 54. 10.5 Graphing Other Trigonometric Functions • Try 560#7 Graph 𝑔 𝑥 = 1 2 tan 𝜋𝑥
  • 55. 10.5 Graphing Other Trigonometric Functions • 𝑦 = csc 𝑥 • Period = 2π • 𝑇 = 2𝜋 𝑏 • Asymptotes where sine = 0 • 0, 𝜋, 2𝜋
  • 56. 10.5 Graphing Other Trigonometric Functions • 𝑦 = sec 𝑥 • Period = 2π • 𝑇 = 2𝜋 𝑏 • Asymptotes where cosine = 0 • 𝜋 2 , 3𝜋 2
  • 57. 10.5 Graphing Other Trigonometric Functions • Graph 𝑦 = 2 csc 𝑥 + 𝜋 2
  • 58. 10.5 Graphing Other Trigonometric Functions • Try 560#15 Graph 𝑔 𝑥 = csc 4𝑥
  • 59. 10.6 Modeling with Trigonometric Functions After this lesson… • I can write and graph trigonometric functions using frequency. • I can write trigonometric functions for a given graph. • I can find a trigonometric model for a set of data using technology. 59
  • 60. 10.6 Modeling with Trigonometric Functions • Trigonometric functions are periodic • Useful for modeling oscillating motions or repeated patterns • Period (T) • Time of 1 cycles • Unit: s/cycle = s • Frequency (f) • Cycles per 1 second • Unit: cycles/s = Hz 60
  • 61. 10.6 Modeling with Trigonometric Functions • Find the frequency 𝑦 = 2 cos 3𝑥 • Try 568#3 𝑦 = sin 3𝜋𝑥 61
  • 62. 10.6 Modeling with Trigonometric Functions • Write Trigonometric Models • Find the amplitude • Find the period • If the situation starts at zero, use sine • If starts increasing + • If starts decreasing − • If the situation starts at a maximum or minimum use cosine • If starts at max + • If starts at min − 62
  • 63. 10.6 Modeling with Trigonometric Functions • An audiometer produces a pure tone with a frequency f of 1000 hertz (cycles per second). The maximum pressure P produced by the tone is 20 millipascals. Write a sine model that gives the pressure P as a function of the time t (in seconds). 63
  • 64. 10.6 Modeling with Trigonometric Functions • Try 568#9 Frequency is measured in hertz, or cycles per second. The lowest frequency of sounds that can be heard by humans is 20 hertz. The maximum pressure P produced from a sound with a frequency of 20 hertz is 0.02 millipascal. Write a sine model that gives the pressure P as a function of the time t (in seconds). 64
  • 65. 10.6 Modeling with Trigonometric Functions • Write a function for the sinusoid shown. 65
  • 66. 10.6 Modeling with Trigonometric Functions • Try 569#11 Write a function for the sinusoid shown. 66
  • 67. 10.6 Modeling with Trigonometric Functions • Two people swing jump ropes. The highest point of the middle of each rope is 80 inches above the ground and the lowest point is 2 inches above the ground. Each rope makes 2 revolutions per second. Write a model for the height h (in inches) of one of the ropes as a function of the time t (in seconds) given that the rope is at its lowest point when t = 0. 67
  • 68. 10.6 Modeling with Trigonometric Functions • The tables show the average monthly low temperatures D (in degrees Fahrenheit) in Erie, Pennsylvania, where t = 1 represents January. Write a model that gives D as a function of t and interpret the period of its graph. Use technology. • Press STAT, Edit… enter points • Press STAT  CALC, SinReg 68
  • 69. 10.6 Modeling with Trigonometric Functions • Try 569#19 The tables show the average monthly low temperatures D (in degrees Fahrenheit) in Las Vegas, Nevada, where t = 1 represents January. Write a model that gives D as a function of t and interpret the period of its graph. Use technology. 69
  • 70. 10.7 Using Trigonometric Identities After this lesson… • I can evaluate trigonometric functions using trigonometric identities. • I can simplify trigonometric expressions using trigonometric identities. • I can verify trigonometric identities. 70
  • 71. 10.7 Using Trigonometric Identities • On unit circle • 𝑥 = cos 𝜃 • 𝑦 = sin 𝜃 • 𝑥2 + 𝑦2 = 1 • cos2 𝜃 + sin2 𝜃 = 1 • Trigonometric Identity • Statement showing relationship between two quantities that are always = 71
  • 72. 10.7 Using Trigonometric Identities • Reciprocal Identities • sin 𝑢 = 1 csc 𝑢 csc 𝑢 = 1 sin 𝑢 • cos 𝑢 = 1 sec 𝑢 sec 𝑢 = 1 cos 𝑢 • tan 𝑢 = 1 cot 𝑢 cot 𝑢 = 1 tan 𝑢 • Quotient Identities • tan 𝑢 = sin 𝑢 cos 𝑢 cot 𝑢 = cos 𝑢 sin 𝑢 • Pythagorean Identities • sin2 𝑢 + cos2 𝑢 = 1 • tan2 𝑢 + 1 = sec2 𝑢 • 1 + cot2 𝑢 = csc2 𝑢
  • 73. 10.7 Using Trigonometric Identities • Even/Odd Identities • cos −𝑢 = cos 𝑢 • sec −𝑢 = sec 𝑢 • sin −𝑢 = − sin 𝑢 • tan −𝑢 = − tan 𝑢 • csc −𝑢 = − csc 𝑢 • cot −𝑢 = − cot 𝑢 • Cofunction Identities • sin 𝜋 2 − 𝑢 = cos 𝑢 • cos 𝜋 2 − 𝑢 = sin 𝑢 • tan 𝜋 2 − 𝑢 = cot 𝑢 • cot 𝜋 2 − 𝑢 = tan 𝑢 • sec 𝜋 2 − 𝑢 = csc 𝑢 • csc 𝜋 2 − 𝑢 = sec 𝑢
  • 74. 10.7 Using Trigonometric Identities • Given that sin 𝜃 = − 5 13 and 𝜋 < 𝜃 < 3𝜋 2 , find the values of the other five trigonometric functions of θ. • Try 575#3 Given that tan 𝜃 = − 3 7 and 𝜋 2 < 𝜃 < 𝜋, find the values of the other five trigonometric functions of θ. 74
  • 75. 10.7 Using Trigonometric Identities • Simplify (1 + cos θ )(1 − cos θ ) • Try 575#9 sin 𝑥 cot 𝑥 75
  • 76. 10.7 Using Trigonometric Identities • Verify Trigonometric Identities • Show that trig identities are true by turning one side into the other side • Guidelines 1. Work with 1 side at a time. Start with the more complicated side. 2. Try factor, add fractions, square a binomial, etc. 3. Use fundamental identities 4. If the above doesn’t work, try rewriting in sines and cosines 5. Try something!
  • 77. 10.7 Using Trigonometric Identities • Verify 1 sec 𝜃−tan 𝜃 = sec 𝜃 + tan 𝜃 • Try 575#21 cos 𝜋 2 − 𝑥 cot 𝑥 = cos 𝑥 77
  • 78. 10.8 Using Sum and Difference Formulas After this lesson… • I can evaluate trigonometric expressions using sum and difference formulas. • I can simplify trigonometric expressions using sum and difference formulas. • I can solve trigonometric equations using sum and difference formulas. 78
  • 79. 10.8 Using Sum and Difference Formulas • Sum and Difference Formulas • sin 𝑢 + 𝑣 = sin 𝑢 cos 𝑣 + cos 𝑢 sin 𝑣 • sin 𝑢 − 𝑣 = sin 𝑢 cos 𝑣 − cos 𝑢 sin 𝑣 • cos 𝑢 + 𝑣 = cos 𝑢 cos 𝑣 − sin 𝑢 sin 𝑣 • cos 𝑢 − 𝑣 = cos 𝑢 cos 𝑣 + sin 𝑢 sin 𝑣 • tan 𝑢 + 𝑣 = tan 𝑢+tan 𝑣 1−tan 𝑢 tan 𝑣 • tan 𝑢 − 𝑣 = tan 𝑢−tan 𝑣 1+tan 𝑢 tan 𝑣
  • 80. 10.8 Using Sum and Difference Formulas • Find the exact value of sin 75°. • Try 581#1 tan −15° 80
  • 81. 10.8 Using Sum and Difference Formulas • Find cos(a − b) given that cos 𝑎 = − 12 13 with 𝜋 2 < 𝑎 < 𝜋 and sin 𝑏 = 3 5 with 0 < 𝑏 < 𝜋 2 . • Try 581#9 Find sin 𝑎 + 𝑏 given that cos 𝑎 = 4 5 with 0 < 𝑎 < 𝜋 2 and sin 𝑏 = − 15 17 with 3𝜋 2 < 𝑏 < 2𝜋. 81
  • 82. 10.8 Using Sum and Difference Formulas • Simplify the expression cos(x − π). • Try 581#17 cos 𝑥 + 2𝜋 82
  • 83. 10.8 Using Sum and Difference Formulas • Solve cos 𝑥 + 𝜋 4 + cos 𝑥 − 𝜋 4 = 1 for 0 ≤ 𝑥 < 𝜋. • Try 581#23 Solve on 0 ≤ 𝑥 < 2𝜋 2 sin 𝑥 − 1 = 0 83

Editor's Notes

  1. 65°+360°=425°; 65°-360°=-295° 300°+360°=660°; 300°-360°=-60° #3: −900°+360°=−540°; −540°+360°=−180°; −180°+360°=180° #5: −125°+360°=235°; −125°−360°=−485°
  2. Probably only need to memorize 1st quadrant
  3. 135° 𝜋 180 ° = 3𝜋 4 −50° 𝜋 180 ° =− 5𝜋 18 5𝜋 4 180° 𝜋 =225° #9 40° 𝜋 180° = 2𝜋 9
  4. 𝜃= 𝜋 2 𝑠=𝑟𝜃 𝑠=220 𝜋 2 =110𝜋≈346 𝐴= 1 2 𝑟 2 𝜃 𝐴= 1 2 220 2 𝜋 2 =12100𝜋≈38013 #29 Convert to radians. 100° 𝜋 180° = 5𝜋 9 𝐴= 1 2 𝑟 2 𝜃− 1 2 𝑟 2 𝜃 𝐴= 1 2 2 2 5𝜋 9 − 1 2 1.5 2 5𝜋 9 ≈1.53 𝑚 𝑖 2
  5. Use Pythagorean Theorem to find hypotenuse 3 2 + 4 2 =ℎ𝑦 𝑝 2 ℎ𝑦𝑝=5 sin 𝜃 = 3 5 cos 𝜃 = 4 5 tan 𝜃 = 3 4 csc 𝜃 = 5 3 sec 𝜃 = 5 4 cot 𝜃 = 4 3 Use Pythagorean Theorem to find opposite 10 2 + 𝑜𝑝𝑝 2 = 18 2 100+ 𝑜𝑝𝑝 2 =324 𝑜𝑝𝑝 2 =224 𝑜𝑝𝑝=4 14 sin 𝜃 = 4 14 18 = 2 14 9 cos 𝜃 = 10 18 = 5 9 tan 𝜃 = 4 14 10 = 2 14 5 csc 𝜃 = 9 2 14 = 9 14 28 sec 𝜃 = 9 5 cot 𝜃 = 5 2 14 = 5 14 28
  6. Draw triangle and use pythagorean theorem to find opposite side 𝑎𝑑𝑗=7 ℎ𝑦𝑝=10 7 2 + opp 2 = 10 2 𝑜𝑝𝑝 2 =51 𝑜𝑝𝑝= 51 sin 𝜃 = 51 10 Draw triangle and use Pythagorean theorem to find adjacent side 𝑜𝑝𝑝=7, ℎ𝑦𝑝=11 7 2 +𝑎𝑑 𝑗 2 = 11 2 49+𝑎𝑑 𝑗 2 =121 𝑎𝑑 𝑗 2 =72 𝑎𝑑𝑗=6 2 sin 𝜃 = 7 11 cos 𝜃 = 6 2 11 tan 𝜃 = 7 6 2 = 7 2 12 csc 𝜃 = 11 7 sec 𝜃 = 11 6 2 = 11 2 12 cot 𝜃 = 6 2 7
  7. These triangles can be used to find sin, cos, tan of 30, 60, and 45 degree angles exactly.
  8. A = 90° - 60° = 30° From special rt triangle; tan 60°= 𝑏 7 = 3 1 →𝑏=7 3 cos 60° = 7 𝑐 = 1 2 →𝑐=14 B = 90° - 32° = 58° tan 32° = 𝑎 10 →𝑎=10 tan 32° ≈6.25 cos 32° = 10 𝑐 →𝑐⋅ cos 32° =10→𝑐= 10 cos 32° ≈11.79 #33: B = 90° - 43° = 47° tan 43° = 𝑎 31 →𝑎=31 tan 43° ≈28.91 cos 43° = 31 𝑐 →𝑐· cos 43° =31→𝑐= 31 cos 43° ≈42.39
  9. cos 76° = 2 mi 𝑦 𝑦⋅ cos 76° =2 mi 𝑦= 2 mi cos 76° ≈8.27 mi #37: tan 79° = 𝑤 100 𝑤=100 tan 79° ≈514 𝑚
  10. 𝑟 2 = 𝑥 2 + 𝑦 2 = 3 2 + −3 2 𝑟= 18 =3 2 sin 𝜃 =− 3 3 2 =− 2 2 cos 𝜃 = 3 3 2 = 2 2 tan 𝜃 =− 3 3 =−1 csc 𝜃 =− 2 sec 𝜃 = 2 cot 𝜃 =−1 𝑟 2 = −6 2 + −8 2 =100 𝑟=10 sin 𝜃 =− 4 5 cos 𝜃 =− 3 5 tan 𝜃 = 4 3 csc 𝜃 =− 5 4 sec 𝜃 =− 5 3 cot 𝜃 = 3 4 sin 𝜃 =0 cos 𝜃 =−1 tan 𝜃 =0 csc 𝜃 =𝑢𝑛𝑑 sec 𝜃 =−1 cot 𝜃 =𝑢𝑛𝑑
  11. 𝑟 2 = 𝑥 2 + 𝑦 2 = 3 2 + −3 2 𝑟= 18 =3 2 sin 𝜃 =− 3 3 2 =− 2 2 cos 𝜃 = 3 3 2 = 2 2 tan 𝜃 =− 3 3 =−1 csc 𝜃 =− 2 sec 𝜃 = 2 cot 𝜃 =−1 𝑟 2 = −8 2 + 15 2 =289 𝑟=17 sin 𝜃 = 15 17 cos 𝜃 =− 8 17 tan 𝜃 =− 15 8 csc 𝜃 = 17 15 sec 𝜃 =− 17 8 cot 𝜃 =− 8 15 sin 𝜃 =0 cos 𝜃 =−1 tan 𝜃 =0 csc 𝜃 =𝑢𝑛𝑑 sec 𝜃 =−1 cot 𝜃 =𝑢𝑛𝑑
  12. This gives what is positive. The reciprocal functions are the same (csc is with sin, etc.) Way to remember “All Students Take Calculus”
  13. 180°−150°=30° #15 23𝜋 4 −2𝜋= 15𝜋 4 → 15𝜋 4 −2𝜋= 7𝜋 4 2𝜋− 7𝜋 4 = 𝜋 4
  14. Reference angle is 60° In quadrant IV (cos positive) Use 30°-60°-90° triangle cos −60° = 1 2 #25 Reference angle is 30° In quadrant III (sin negative) Use 30°-60°-90° sin −150° =− sin 30° =− 1 2
  15. 𝑑= 𝑣 2 32 sin 2𝜃 𝑑= 53 2 32 sin 2 8° =24.2 𝑓𝑡
  16. Point out Amplitude period key points
  17. c is like h d is like k
  18. amp: 5; period: 2π #7 Amp: 4; period: π
  19. Same as sine, but amp = 2
  20. Period 𝑇= 2𝜋 𝑏 𝑇= 2𝜋 2𝜋 =1 Amp = 1
  21. 𝑎=2 𝑏=1 →𝑇=2𝜋 ℎ= 𝜋 2 𝑡𝑜 𝑟𝑖𝑔ℎ𝑡 Draw 2 sin 𝑥 first and then do the phase shift
  22. 𝑎=−4=𝑎𝑚𝑝 𝑏=1→𝑇= 2𝜋 𝑏 → 2𝜋 1 =2𝜋 ℎ=− 𝜋 4 Shift left π/4 𝑘=−1 Shift down 1 Graph −4 sin 𝑥 first labeling the key points with a period of 2π Reflect over the x-axis because a is negative Shift left π/4 and down 1
  23. 𝑏= 1 4 𝑇= 𝜋 𝑏 = 𝜋 1 4 =4𝜋 𝑎=1
  24. 𝑏=𝜋 𝑇= 𝜋 𝑏 = 𝜋 𝜋 =1 𝑎= 1 2
  25. 𝑎=2 𝑏=1 𝑇= 2𝜋 𝑏 = 2𝜋 1 =2𝜋 𝑐=− 𝜋 2 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑠ℎ𝑖𝑓𝑡;ℎ=− 𝜋 2 𝑘=0 Start by graphing 2 sin 𝑥 Then shift left 𝜋 2 Then draw asymptotes at the x-intercepts Then draw csc graph
  26. 𝑎=1 𝑏=4 𝑇= 2𝜋 𝑏 = 2𝜋 4 = 𝜋 2 Start by graphing sin 4𝑥 Then draw asymptotes at the x-intercepts Then draw csc graph
  27. Period and frequency are reciprocals
  28. Find period. 𝑇= 2𝜋 𝑏 = 2𝜋 3 𝑓= 1 𝑇 = 1 2𝜋 3 = 3 2𝜋 #3 𝐹= 2𝜋 𝑏 = 2𝜋 3𝜋 = 2 3 𝑓= 1 𝑇 = 3 2
  29. Sound is produced by compressing air 𝑓=1000 𝐻𝑧 𝑇= 1 𝑓 = 2𝜋 𝑏 = 1 1000 →𝑏=2000𝜋 𝑎=20 𝑚𝑃𝑎 𝑃=20 sin 2000𝜋𝑡
  30. 𝑓= 1 𝑇 =20→𝑇= 1 20 = 2𝜋 𝑏 →𝑏=40𝜋 𝑎=0.02 𝑦=0.02 sin 40𝜋𝑡
  31. y-intercept is a max, so use cosine and h = 0 Midline is 𝑦= 13+ −5 2 =4=𝑘 Amplitude is 13−4=9=𝑎 Period is 𝑇= 𝜋 4 = 2𝜋 𝑏 →𝜋𝑏=8𝜋→𝑏=8 𝑦=𝑎 cos 𝑏 𝑥−ℎ +𝑘 𝑦=9 cos 8 𝑥−0 +4 𝑦=9 cos 8𝑥 +4
  32. y-intercept is a 0, so use sine and h = 0 Midline is 𝑦=0=𝑘 Amplitude is 3−0=3=𝑎 Period is 𝑇=𝜋= 2𝜋 𝑏 →𝜋𝑏=2𝜋→𝑏=2 𝑦=𝑎 sin 𝑏 𝑥−ℎ +𝑘 𝑦=3 sin 2 𝑥−0 +0 𝑦=3 sin 2𝑥
  33. Midline: 80+2 2 =41=𝑘 Amplitude: 80−41=39=𝑎 Frequency: 𝑓=2= 1 𝑇 →𝑇= 1 2 = 2𝜋 𝑏 →𝑏=4π Lowest point at t = 0, so use cosine with −a, 𝑎=−39 𝑦=𝑎 cos 𝑏 𝑥−ℎ +𝑘 ℎ 𝑡 =−39 cos 4𝜋𝑡 +41
  34. Use a graphing calculator D = 21.3 sin(0.52t − 2.3) + 42 The period 12 makes sense because there are 12 months in a year, and you expect this pattern to continue in the years to follow.
  35. Use a graphing calculator D = 22.90 sin(0.542t − 2.25) + 80.9 The period 12 makes sense because there are 12 months in a year, and you expect this pattern to continue in the years to follow.
  36. Colored ones should be memorized.
  37. Quadrant III, so x<0 and y<0 sin 2 𝜃 + cos 2 𝜃 =1→ − 5 13 2 + cos 2 𝜃 =1→ cos 2 𝜃 =1− 25 169 = 144 169 → cos 𝜃 =− 12 13 tan 𝜃 = sin 𝜃 cos 𝜃 = − 5 13 − 12 13 = 5 12 csc 𝜃 = 1 sin 𝜃 =− 13 5 sec 𝜃 = 1 cos 𝜃 =− 13 12 cot 𝜃 = 1 tan 𝜃 = 12 5 #3: Quadrant II, so sin +, cos −, tan − 1+ tan 2 𝜃 = sec 2 𝜃 →1+ − 3 7 2 = sec 2 𝜃 → 58 49 = sec 2 𝜃 → sec 𝜃 =− 58 7 cos 𝜃 = 1 sec 𝜃 = 1 − 58 7 =− 7 58 58 tan 𝜃 = sin 𝜃 cos 𝜃 →− 3 7 = sin 𝜃 − 7 58 58 → sin 𝜃 = 3 58 58 csc 𝜃 = 1 sin 𝜃 = 58 3 cot 𝜃 = 1 tan 𝜃 =− 7 3
  38. 1+ cos 𝜃 1− cos 𝜃 1− cos 2 𝜃 (because sin 2 𝜃 + cos 2 𝜃 =1, then cos 2 𝜃 =1− sin 2 𝜃 ) 1− 1− sin 2 𝜃 sin 2 𝜃 #9 sin 𝑥 cot 𝑥 sin 𝑥 cos 𝑥 sin 𝑥 cos 𝑥
  39. 1 sec 𝜃 − tan 𝜃 1 sec 𝜃 − tan 𝜃 · sec 𝜃 + tan 𝜃 sec 𝜃 + tan 𝜃 sec 𝜃 + tan 𝜃 sec 2 𝜃 − tan 2 𝜃 Use sec 2 𝜃 =1+ tan 2 𝜃 sec 𝜃 + tan 𝜃 1+ tan 2 𝜃 − tan 2 𝜃 sec 𝜃 + tan 𝜃 #21 cos 𝜋 2 −𝑥 cot 𝑥 sin 𝑥 cot 𝑥 sin 𝑥 cos 𝑥 sin 𝑥 cos 𝑥
  40. sin 75° = sin 30°+45° = sin 30° cos 45° + cos 30° sin 45° = 1 2 · 2 2 + 3 2 · 2 2 = 2 4 + 6 4 = 2 + 6 4 #1 tan −15° = tan 30°−45° = tan 30° − tan 45° 1+ tan 30° · tan 45° = 3 3 −1 1+ 3 3 ·1 = −1+ 3 3 1+ 3 3 = −3+ 3 3+ 3 = −3+ 3 3+ 3 · 3− 3 3− 3 = −9−6 3 −3 9−3 = −12−6 3 6 =−2− 3
  41. Find sin a sin 2 𝑎 + cos 2 𝑎 =1 sin 2 𝑎 + − 12 13 2 =1→ sin 𝑎 = 5 13 Find cos b 3 5 2 + cos 2 𝑏 =1→ cos 𝑏 = 4 5 cos 𝑎−𝑏 = cos 𝑎 cos 𝑏 + sin 𝑎 sin 𝑏 = − 12 13 4 5 + 5 13 3 5 =− 33 65 #9 Find sin a sin 2 𝑎 + cos 2 𝑎 =1 sin 2 𝑎 + 4 5 2 =1→ sin 𝑎 = 3 5 Find cos b − 15 17 2 + cos 2 𝑏 =1→ cos 𝑏 = 8 17 sin 𝑎+𝑏 = sin 𝑎 cos 𝑏 + cos 𝑎 sin 𝑏 = 3 5 8 17 + 4 5 − 15 17 =− 36 85
  42. cos 𝑥−𝜋 = cos 𝑥 cos 𝜋 + sin 𝑥 sin 𝜋 = cos 𝑥 −1 + sin 𝑥 0 =− cos 𝑥 #17 cos (𝑥+2𝜋) = cos 𝑥 cos 2𝜋 − sin 𝑥 sin 2𝜋 = cos 𝑥 1 − sin 𝑥 0 = cos 𝑥
  43. For both terms a = x and b = π/4 cos 𝑥+ 𝜋 4 + cos 𝑥− 𝜋 4 =1 cos 𝑥 cos 𝜋 4 − sin 𝑥 sin 𝜋 4 + cos 𝑥 cos 𝜋 4 + sin 𝑥 sin 𝜋 4 =1 2 2 cos 𝑥 − 2 2 sin 𝑥 + 2 2 cos 𝑥 + 2 2 sin 𝑥 =1 2 2 2 cos 𝑥 =1 2 cos 𝑥 =1 cos 𝑥 = 1 2 = 2 2 From unit circle 𝑥= π 4 #23 2 sin 𝑥 −1=0 2 sin 𝑥 =1 sin 𝑥 = 1 2 From unit circle 𝑥= 𝜋 6 , 5𝜋 6