AC Waveforms Lesson Plan
Course Title: Electronics
Session Title: AC Waveforms
Lesson Duration: 2-4 hours
[Lesson length is subjective and will vary from instructor to instructor]
Performance Objective:
Upon completion of this assignment, the student will be able to explain sinusoidal and
nonsinusoidal waveforms, define the waveform terminology, and calculate various values
related to waveforms.
Specific Objectives:
• Understand how a sine wave of alternating voltage is generated.
• Define frequency and period and list the units of each.
• Define the following values for a sine wave: peak, peak-to-peak, root means-square,
average, and instantaneous.
• Understand the concept of phase angles.
• Explain the three ways to express the amplitude of a sinusoidal waveform and the
relationship between them.
• Explain the importance of the .707 constant and how it is derived.
• Calculate the wavelength when the frequency is known.
• Calculate the rms, average, and peak-to-peak values of a sine wave when the peak value is
known.
• Calculate the instantaneous value of a sine wave.
• Convert peak, peak-to-peak, and rms voltage and current values from one value to another.
• Explain the sine, cosine, and tangent trigonometric functions.
• Calculate the value of the sine of any sine of any angle between 0°and 360°.
• Understand the makeup of a nonsinusoidal waveform.
Preparation
TEKS Correlations:
This lesson, as published, correlates to the following TEKS. Any changes/alterations to the
activities may result in the elimination of any or all of the TEKS listed.
Electronics:
•
130.368 (c)(5)(C)(D)(E)
...demonstrate knowledge of the fundamentals of electronics theory;
...perform electrical-electronic troubleshooting assignments;
...develop knowledge of voltage regulation devices.
•
130.368 (c)(6)(A)(B)(C)(D)
...measure and calculate resistance, current, voltage, and power in series, parallel, and
complex circuits;
...apply electronic theory to generators, electric motors, and transformers;
Electronics: AC Waveforms Plan
1
...design analog and digital circuits using common components; and
...demonstrate knowledge of common devices in optoelectronics.
Interdisciplinary Correlations:
Computer Technologies:
•
125.46(c)(3)(C)
…demonstrate knowledge of digital and analog electronics theory;
Telecommunication Services:
•
125.47(c)(3)(B)
…demonstrate knowledge of digital and analog electronics theory;
English:
110.42(b)(6) – Vocabulary development
•
110.42(b)(6)(A)(B)(E)
…expand vocabulary through wide reading, listening, and discussing;
…rely on context to determine meanings of words and phrases such as figurative
language, idioms, multiple meaning words, and technical vocabulary;
…use reference materials ...to determine precise meaning and usage;
• 110.42(b)(7)(B)
…draw upon his/her background to provide connections to texts;
110.42(b)(20) – Viewing/representing/analysis
• 110.42(b)(20)(B)
…deconstruct media to get the main idea of the message's content;
Algebra I and II:
•
111.32(a)(5)
... use a variety of representations (concrete, pictorial, numerical, symbolic, graphical
and verbal), tools, and technology (including, but not limited to, calculators with graphing
capabilities, data collection devices, and computers) to model mathematical situations
to solve meaningful problems.
Geometry:
•
111.34(a)(5)
…use a variety of representations (concrete, pictorial, numerical, symbolic, graphical,
and verbal), tools, and technology (including, but not limited to, calculators with graphing
capabilities, data collection devices, and computers) to solve meaningful problems by
representing and transforming figures and analyzing relationships.
Electronics: AC Waveforms Plan
2
•
111.34(b)(11)(C)
...develop, apply, and justify triangle similarity relationships, such as right triangle ratios,
trigonometric ratios, and Pythagorean triples using a variety of methods;
Precalculus:
•
111.35(b)(1)
…use functions as well as symbolic reasoning to represent and connect ideas in
geometry, probability, statistics, trigonometry, and calculus and to model physical
situations...use a variety of representations (concrete, pictorial, numerical, symbolic,
graphical and verbal), tools, and technology (including, but not limited to, calculators with
graphing capabilities, data collection devices, and computers) to model functions and
equations and solve real-life problems.
•
111.35(c)(1)
…defines functions, describes characteristics of functions, and translates among verbal,
numerical, graphical, and symbolic representations of functions, including polynomial,
rational, power (including radical), exponential, logarithmic, trigonometric, and
piecewise-defined functions.
•
111.35(c)(3)(B)(E)
…use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to
model real-life data;
…solve problems from physical situations using trigonometry, including the use of Law
of Sines, Law of Cosines, and area formulas and incorporate radian measure where
needed;
Mathematical Models with Applications:
•
111.36(c)(8)(B)
…use trigonometric ratios and functions available through technology to calculate
distances and model periodic motion;
Instructor/Trainer
References:
1. Basic Electronics by Grob / Schultz Publisher: Glencoe/McGraw-Hill
2. Electricity & Electronics by Gerrish/Dugger/Roberts Publisher: Goodheart-Willcox
Company
Instructional Aids:
1. AC Waveforms (without Exercise Key) PowerPoint Presentation
2. AC Waveforms (without Exercise Key) PowerPoint Presentation - Slides
3. AC Waveforms (without Exercise Key) PowerPoint Presentation - Handouts
4. AC Waveforms (without Exercise Key) PowerPoint Presentation - Notes Pages
5. AC Waveforms (with Exercise Key) PowerPoint Presentation
6. AC Waveforms (with Exercise Key) PowerPoint Presentation - Slides
7. AC Waveforms (with Exercise Key) PowerPoint Presentation - Handouts
8. AC Waveforms (with Exercise Key) PowerPoint Presentation - Notes Pages
9. AC Waveforms Student Study Guide
Electronics: AC Waveforms Plan
3
10. AC Waveforms Student Study Guide Key
11. AC Waveforms Exam
12. AC Waveforms Exam key
Materials Needed:
None
Equipment Needed:
1. Projection system to display the PowerPoint Presentation
2. Calculator (prefer Texas Instrument models TI-30Xa or TI-36
Learner Preparation:
Proper use of a scientific calculator.
Introduction
Introduction (LSI Quadrant I):
This lesson discusses in detail the amplitude descriptions of a sinusoidal waveform. Also, the
time and frequency measurement of a waveform are reviewed, and an introduction to the
trigonometric function will be presented.
Outline
•
•
NOTE: There are two sets of PowerPoint materials.
• If you DO NOT want to show or provide the answers to the Student Guide exercises with
the PowerPoint materials, instructors can use the PowerPoint presentation, slides,
handout, and note pages (without Exercise Key) in conjunction with the following outline.
• If you want to show or provide the answers to the Student Guide exercises with the
PowerPoint materials, instructors can use the PowerPoint presentation, slides, handout,
and note pages (with Exercise Key) in conjunction with the following outline.
NOTE: Instructors can also use the Student Guide and Guide Key in conjunction with the
following outline
Outline (LSI Quadrant II):
Instructor Notes:
1. Overview
• wave
• waveform
•
•
PowerPoint slide 6
Student Guide page 1
2. Define the Waveform Terminology
• frequency
• period
• amplitude
• sinusoidal waveform
• nonsinusoidal waveform
• instantaneous
• average
• phase angles
•
•
PowerPoint slides 7-8
Student Guide page 1
Electronics: AC Waveforms Plan
4
3. Discuss Each Aspect of the Waveform Amplitude
Specification
• peak
• peak-to-peak
a. sinusoidal waveform
b. nonsinusoidal waveform
• root-mean-square (rms)
•
•
PowerPoint slides 9-11
Student Guide pages 1-4
4. Discuss RMS relations to DC Heating Effect
• Erms = 0.707 Epeak
• Irms = 0.707 Ipeak
• Determining 0.707 constant
•
•
PowerPoint slides 12-14
Student Guide page 4
5. Use Formulas in AC Waveform
• peak to peak
• peak
• root – mean - square
• instantaneous
• average
• .637
• .707
• 2
•
•
•
•
PowerPoint slides 15-24
PowerPoint (with Exercise
Key) slide 33
Student Guide pages 4-5
Student Guide Key page 5
6. Explain Importance of Trigonometry to Waveforms
•
PowerPoint slide 25
7. Define the Trigonometric Terminology
• Right-Triangle
• Opposite
• Hypotenuse
• Sine
• Cosine
• Tangent
•
•
PowerPoint slides 26-30
Student Guide pages 6-7
8. Use Ratios and Formulas in Right-Side Trigonometric
Functions
• Opposite
• Hypotenuse
• Sine
• Cosine
• Tangent
•
•
PowerPoint slide 31
PowerPoint (with Exercise
Key) slides 34-38
Student Guide page 8
Student Guide Key pages
8-10
•
•
Application
Guided Practice (LSI Quadrant III):
1. Instructor works through problems as students follow along and record solution and take
Electronics: AC Waveforms Plan
5
notes.
2. Instructor observes students using the calculator.
Independent Practice (LSI Quadrant III):
Students work independently to complete student worksheets in the Student Guide.
Summary
Review (LSI Quadrants I and IV):
Restate lesson objectives or have students recall lesson objectives.
Evaluation
Informal Assessment (LSI Quadrant III):
1. Instructor monitors individual/group progress as students work on activities. Instructor
provides individual help/redirection as needed.
2. Instructor can use the Student Guide Key to assign a grade for the student worksheets.
Formal Assessment (LSI Quadrant III, IV):
Student will take a math test and fill in the blanks. Use AC Waveform Exam and Exam Key.
Extension
Extension/Enrichment (LSI Quadrant IV):
• Students make a list of common waves and categorize them into sinusoidal and
nonsinusoidal waveforms (radio, TV, sound, heat, ocean, etc.)
• Students research the use of oscilloscopes.
• Students use an oscilloscope.
Electronics: AC Waveforms Plan
6
AC Waveforms
Student Study Guide
Waveform
A wave is a disturbance traveling through a medium. A waveform is a graphic representation of
a wave. Like a wave, a waveform depends both on movement and on time. The ripple on the
surface of a pond is a movement of water in time. Wave shapes tell you a great deal about the
signal. Any time you see a change in the vertical dimension of a signal, you know that this
amplitude change represents a change in voltage. But wave shapes alone are not the whole
story. To completely describe a waveform, you’ll want to find its particular parameters.
Depending on the signal, these parameters might be frequency, period, amplitude, width, rise
time, or phase.
Frequency
The frequency of a waveform is the number of cycles of the waveform which occur in one
second of time. Common unit of measurement is hertz (Hz).
Period
The period of a waveform, which sometimes is called its time, is the time required to complete
one cycle of a waveform. It is measured in units of seconds, such as seconds, tenths of
seconds, millisecond, or microseconds.
Figure 2.1 Sample of Waveform
If a waveform is to be properly described in terms of its period or frequency, it must be a
repetitious waveform. A repetitious waveform is one in which each following cycle is identical to
the previous cycle.
Waveform Amplitude Specifications
In addition to frequency and period values, a third major specification of a waveform is the
amplitude or height of the wave. There are three possible ways to express the amplitude of a
sinusoidal waveform: peak, peak-to-peak, and root-mean-square (rms).
Electronics: AC Waveforms Plan
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Peak: The peak amplitude of a sinusoidal waveform is the maximum positive or negative
deviation of a waveform from its zero reference level. The sinusoidal waveform is a symmetrical
waveform, so the positive peak value is the same as the negative peak value as shown in figure
2.2. If the positive peak has a value of 10 volts, then the negative peak will also have a value of
10 volts. When measuring the peak value of a waveform, either positive or negative peaks can
be used.
Figure 2.2 Positive Peak & Negative Peak Value
Peak–to-Peak: The peak-to-peak amplitude is simply a measurement of the amplitude of a
waveform taken from its positive peak to its negative peak as shown in figure 2.3.
Figure 2.3 Peak-to-Peak Amplitude
Electronics: AC Waveforms Plan
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For sinusoidal waveform, if the positive peak value is 10 volts in magnitude, then the negative
peak value of the same waveform is also 10 volts. Measuring from peak-to-peak, there is a total
of 20 volts. Therefore, the value of the sinusoidal waveform in figure 2.2 can be specified as
either 10 volts peak or 20 volts peak-to-peak.
For the non-sinusoidal waveform shown in figure 2.4, the peak-to-peak value of the voltage
can be determined by adding the magnitude of the positive and the negative peak. In this
example, the peak-to-peak amplitude is 18 volts plus 2 volts for a total of 20 volts, peak-to-peak.
Root-Mean-Square: The third specification for ac waveform is called root-mean-square
abbreviated rms. This term allows the comparison of ac and dc circuit values. Root-meansquare values are the most common methods of specifying sinusoidal waveforms. In fact,
almost all ac voltmeter and ammeters are calibrated so that they measure ac values in terms of
rms amplitude.
RMS Relations to DC Heating Effect
The rms value is also known as the effective value and is defined in terms of the equivalent
heating effect of direct current. The rms value of a sinusoidal voltage is equivalent to the value
of a dc voltage which causes an equal amount of heat due to the circuit current flowing through
a resistance. The rms value of a sinusoidal voltage or current waveform is 70.7 percent or 0.707
of its peak amplitude value.
Erms = 0.707 Epeak
Irms = 0.707 Ipeak
A sinusoidal voltage with peak amplitude of 1 volt has the same effect as a dc voltage of 0.707
volts as far as its ability to reproduce the same amount of heat in a resistance. Because the ac
voltage of 1 volt peak or 0.707 volts rms is as effective as a dc voltage of 0.707 volts, the rms
value of voltage is also referred to as the effective value.
Determining the 0.707 Constant
How is the 70.7 percent of peak-value constant derived? Essentially, the words root-meansquare tell how because they define the mathematical procedure used to determine the
constant.
Electronics: AC Waveforms Plan
9
P-P
2
RMS
PK
0.707 PK
INST.
AVG
Sine ° PK
0.637 PK
EXAMPLES
120 VAC = 170 Vpk
FORMULA: PK = RMS ÷ 0.707
120 ÷ 0.707 = 169.7
(round off to 170 Vpk)
18 V @ 72 ° = 19 Vpk
FORMULA: PK = Instantaneous ÷ Sine
18 ÷ 72 Sine = 18.9
(round off to 19)
25 mVpk = 17.7 VAC
FORMULA RMS = 0.707 X PK
0.707 X 25 EE -3 = 17.675 -3
(round off to 17.7 VAC)
350 V @ 23.5° = 30° Vpk
FORMULA: PK = Instantaneous ÷ Sine
350 ÷ 23.5 Sine = 877.7
(round off to 878)
50 mpp = 17.7 Vrms
First Step: Need to find pk
FORMULA: PK = p-p ÷ 2
50 EE -3 ÷ 2 = 25m
Second Step: Find RMS
FORMULA: RMS = 0.707 X PK
0.707 X 25 EE -3 = 17.675 -3
(round off to 17.7 Vrms)
454 V instantaneous with a pk of 908 V 30°
FORMULA: Sine = Instantaneous ÷ PK
454 ÷908 = 0.5 2nd Sine 30°
20 V Average = 22.2 Vrms
= 31.4 Vpk
= 62.8 Vp-p
First Step: Find pk, Formula: Pk = Average ÷ 0.637
= 20 ÷0.637 = 31.39 (round off to 31.4)
Second Step: Find rms, Formula: RMS = 0.707 X PK
= 0.707 X 31.4 = 22.19 (round off to 22.2)
Third Step: Find p-p, Formula: PK = 2 X PK
= 2 X 31.4 = 62.8
Electronics: AC Waveforms Plan
10
#
3.
rms
pk-to-pk
average
113V
________ V @ 90º
96.4
________ V @ 235º
6.
7.
instantaneous
________ ____ V @
72º
200mV
4.
5.
peak
1.5V @ 122º
689µV
________ V @ 35º
Electronics: AC Waveforms Plan
11
The Sine Wave and Sine Trigonometric Function
The term sinusoidal has been used to describe a waveform produced by an ac generator. The
term sinusoidal comes from a trigonometric function called the sine function.
Right-Triangle: Side and Angle Relationships
As you may know, trigonometry is the study of triangles and their relationship. The basic triangle
studied in trigonometry is a right triangle which is a triangle that has a 90◦ angle as one of its
three angles. A 90◦ triangle has a unique set of relationships from which the rules for
trigonometry are derived.
To help distinguish the sides of a right triangle from one another, a name is given to each side.
The sides of the triangle are named with respect to the angle theta. The side of the triangle
across from or opposite to the angle theta is called the opposite side.
The longest side of a right triangle is called the hypotenuse. The remaining side is called the
adjacent side because it lies beside or adjacent to the angle. These three names are commonly
abbreviated to their first initials: O. H and A.
Basic Trigonometric Functions
In trigonometry, these ratios have specific names. The three most commonly-used ratios in the
study of right triangles are called sine, cosine, and tangent. The sine of the angle theta is equal
to the ratio formed by the length of the opposite side divided by the length of the hypotenuse:
sine ө = opposite
hypotenuse
The cosine of the angle theta is equal to the ratio formed by length of the adjacent side divided
by the length of the hypotenuse:
Cosine ө
hypotenuse
The tangent of the angle theta is equal to the ratio formed by length of the opposite side divided
by the length of the adjacent side:
Tangent ө = opposite
Electronics: AC Waveforms Plan
12
Right Triangle
Hypotenuse
Opposite
Formulas
Opposite
Sine
Hypotenuse
Cosine
Hypotenuse
Opposite
Tangent
Electronics: AC Waveforms Plan
13
Triangle #1
Triangle #2
Hypotenuse?
Hypotenuse?
8' Opposite
5.3 Rods
Opposite
Hypotenuse:
Hypotenuse:
Triangle #3
Triangle #4
125 miles
Hypotenuse?
56'
Hypotenuse?
Opposite?
Opposite:
23.2'
Opposite
Electronics: AC Waveforms Plan
14
AC Waveforms
Student Study Guide Key
Waveform
A wave is a disturbance traveling through a medium. A waveform is a graphic representation of
a wave. Like a wave, a waveform depends both on movement and on time. The ripple on the
surface of a pond is a movement of water in time. Wave shapes tell you a great deal about the
signal. Any time you see a change in the vertical dimension of a signal, you know that this
amplitude change represents a change in voltage. But wave shapes alone are not the whole
story. To completely describe a waveform, you’ll want to find its particular parameters.
Depending on the signal, these parameters might be frequency, period, amplitude, width, rise
time, or phase.
Frequency
The frequency of a waveform is the number of cycles of the waveform which occur in one
second of time. Common unit of measurement is hertz (Hz).
Period
The period of a waveform, which sometimes is called its time, is the time required to complete
one cycle of a waveform. It is measured in units of seconds, such as seconds, tenths of
seconds, millisecond, or microseconds.
Figure 2.1 Sample of Waveform
If a waveform is to be properly described in terms of its period or frequency, it must be a
repetitious waveform. A repetitious waveform is one in which each following cycle is identical to
the previous cycle.
Waveform Amplitude Specifications
In addition to frequency and period values, a third major specification of a waveform is the
amplitude or height of the wave. There are three possible ways to express the amplitude of a
sinusoidal waveform: peak, peak-to-peak, and root-mean-square (rms).
Electronics: AC Waveforms Plan
15
Peak: The peak amplitude of a sinusoidal waveform is the maximum positive or negative
deviation of a waveform from its zero reference level. The sinusoidal waveform is a symmetrical
waveform, so the positive peak value is the same as the negative peak value as shown in figure
2.2. If the positive peak has a value of 10 volts, then the negative peak will also have a value of
10 volts. When measuring the peak value of a waveform, either positive or negative peaks can
be used.
Figure 2.2 Positive Peak & Negative Peak Value
Peak–to-Peak: The peak-to-peak amplitude is simply a measurement of the amplitude of a
waveform taken from its positive peak to its negative peak as shown in figure 2.3.
Figure 2.3 Peak-to-Peak Amplitude
Electronics: AC Waveforms Plan
16
For sinusoidal waveform, if the positive peak value is 10 volts in magnitude, then the negative
peak value of the same waveform is also 10 volts. Measuring from peak-to-peak, there is a total
of 20 volts. Therefore, the value of the sinusoidal waveform in figure 2.2 can be specified as
either 10 volts peak or 20 volts peak-to-peak.
For the non-sinusoidal waveform shown in figure 2.4, the peak-to-peak value of the voltage
can be determined by adding the magnitude of the positive and the negative peak. In this
example, the peak-to-peak amplitude is 18 volts plus 2 volts for a total of 20 volts, peak-to-peak.
Root-Mean-Square: The third specification for ac waveform is called root-mean-square
abbreviated rms. This term allows the comparison of ac and dc circuit values. Root-meansquare values are the most common methods of specifying sinusoidal waveforms. In fact,
almost all ac voltmeter and ammeters are calibrated so that they measure ac values in terms of
rms amplitude.
RMS Relations to DC Heating Effect
The rms value is also known as the effective value and is defined in terms of the equivalent
heating effect of direct current. The rms value of a sinusoidal voltage is equivalent to the value
of a dc voltage which causes an equal amount of heat due to the circuit current flowing through
a resistance. The rms value of a sinusoidal voltage or current waveform is 70.7 percent or 0.707
of its peak amplitude value.
Erms = 0.707 Epeak
Irms = 0.707 Ipeak
A sinusoidal voltage with peak amplitude of 1 volt has the same effect as a dc voltage of 0.707
volts as far as its ability to reproduce the same amount of heat in a resistance. Because the ac
voltage of 1 volt peak or 0.707 volts rms is as effective as a dc voltage of 0.707 volts, the rms
value of voltage is also referred to as the effective value.
Determining the 0.707 Constant
How is the 70.7 percent of peak-value constant derived? Essentially, the words root-meansquare tell how because they define the mathematical procedure used to determine the
constant.
Electronics: AC Waveforms Plan
17
P-P
2
RMS
PK
0.707 PK
INST.
AVG.
Sine ° PK
0.637 PK
EXAMPLES
120 VAC = 170 Vpk
FORMULA: PK = RMS ÷ 0.707
120 ÷ 0.707 = 169.7
(round off to 170 Vpk)
18 V @ 72 ° = 19 Vpk
FORMULA: PK = Instantaneous ÷ Sine
18 ÷ 72 Sine = 18.9
(round off to 19)
25 mVpk = 17.7 VAC
FORMULA RMS = 0.707 X PK
0.707 X 25 EE -3 = 17.675 -3
(round off to 17.7 VAC)
350 V @ 23.5° = 30° Vpk
FORMULA: PK = Instantaneous ÷ Sine
350 ÷ 23.5 Sine = 877.7
(round off to 878)
50 mpp = 17.7 Vrms
First Step: Need to find pk
FORMULA: PK = p-p ÷ 2
50 EE -3 ÷ 2 = 25m
Second Step: Find RMS
FORMULA: RMS = 0.707 X PK
0.707 X 25 EE -3 = 17.675 -3
(round off to 17.7 Vrms)
454 V instantaneous with a pk of 908 V 30°
FORMULA: Sine = Instantaneous ÷ PK
454 ÷908 = 0.5 2nd Sine 30°
20 V Average = 22.2 Vrms
= 31.4 Vpk
= 62.8 Vp-p
First Step: Find pk, Formula: Pk = Average ÷ 0.637
= 20 ÷0.637 = 31.39 (round off to 31.4)
Second Step: Find rms, Formula: RMS = 0.707 X PK
= 0.707 X 31.4 = 22.19 (round off to 22.2)
Third Step: Find p-p, Formula: PK = 2 X PK
= 2 X 31.4 = 62.8
Electronics: AC Waveforms Plan
18
#
rms
peak
pk-to-pk
average
instantaneous
8.
200mV
283mV
566mV
180mV
269mV @ 72º
9.
80v
113V
226V
72V
113V @ 90º
10.
96.4
136V
272V
87V
111V @ 235º
11.
1.25V
1.77V
3.54V
1.13V
1.5V @ 122º
12.
764µV
1.08mV
2.16mV
689µV
619µV @ 35º
Electronics: AC Waveforms Plan
19
The Sine Wave and Sine Trigonometric Function
The term sinusoidal has been used to describe a waveform produced by an ac generator. The
term sinusoidal comes from a trigonometric function called the sine function.
Right-Triangle: Side and Angle Relationships
As you may know, trigonometry is the study of triangles and their relationship. The basic triangle
studied in trigonometry is a right triangle which is a triangle that has a 90◦ angle as one of its
three angles. A 90◦ triangle has a unique set of relationships from which the rules for
trigonometry are derived.
To help distinguish the sides of a right triangle from one another, a name is given to each side.
The sides of the triangle are named with respect to the angle theta. The side of the triangle
across from or opposite to the angle theta is called the opposite side.
The longest side of a right triangle is called the hypotenuse. The remaining side is called the
adjacent side because it lies beside or adjacent to the angle. These three names are commonly
abbreviated to their first initials: O. H and A.
Basic Trigonometric Functions
In trigonometry, these ratios have specific names. The three most commonly-used ratios in the
study of right triangles are called sine, cosine, and tangent. The sine of the angle theta is equal
to the ratio formed by the length of the opposite side divided by the length of the hypotenuse:
sine ө = opposite
hypotenuse
The cosine of the angle theta is equal to the ratio formed by length of the adjacent side divided
by the length of the hypotenuse:
Cosine ө
hypotenuse
The tangent of the angle theta is equal to the ratio formed by length of the opposite side divided
by the length of the adjacent side:
Tangent ө = opposite
Electronics: AC Waveforms Plan
20
Right Triangle
Hypotenuse
Opposite
Formulas
Opposite
Sine
Hypotenuse
Cosine
Hypotenuse
Opposite
Tangent
Electronics: AC Waveforms Plan
21
Triangle #1
Triangle #2
Hypotenuse?
Hypotenuse?
8' Opposite
Hypotenuse:
5.3 Rods
Opposite
12.8'
Triangle #3
Hypotenuse:
8.62 Rods
Triangle #4
125 miles
Hypotenuse?
56'
Hypotenuse?
Opposite?
Opposite:
91.7 miles
23.2'
Opposite
51'
NOTE: See next page for detail answers.
Electronics: AC Waveforms Plan
22
Triangle #1
Opposite: 8’
1st step Find the degree angle.
Tangent = Opposite
=8
10
= .8 enter 2nd tan on your calculator
= 38.65980825 round up to 38.66º
= 38.66°
2nd step Change the degree angle to cosine.
Cosine
= ___10’ __ take 38.66° enter cosine on your calculator your answer is
0.780866719
.780866719
= 12.8’
Triangle #2
Hypotenuse?
Opposite: 5.3 rods
1st step Find the degree angle.
Tanget = Opposite
= 5.3 rods
6.8 rods
= 0.779411765 enter 2nd tan on your calculator
= 37.93°
2nd step Change the degree angle to sine.
Hypot = Opposite
Sine
= 5.3 rods
37.93º
enter 37.93, enter sine on the calculator
= ___5.3 rods___
614.6982793-3
= 8.63
Electronics: AC Waveforms Plan
23
Triangle #3
Hypotenuse: 125 miles
Opposite:?
1st step Find the degree angle.
Hypotenuse
= 85 miles
125 miles
= 0.68 enter 2nd function button on calculator enter cosine
= 47.15635696 (round off to 47.16º)
2nd step change the degree angle to sine.
Opposite = Sine x Hypotenuse
47.16 enter sine on calculator
= 733.2553462 -3
= 733.2553462-3 x 125 miles
= 91.65691828 (round off to 91.7)
= 91.7 miles
Triangle #4
Hypotenuse: 56’
Opposite: 23.2’
1st step Find the degree angle.
Sine = _Opposite__
Hypotenuse
= 23.2’
56’
= 414.2857143-3 enter 2nd sin 24.47434256
= 24.47º
2nd step change the degree angle to cosine.
Adjacent = Cosine x Hypotenuse
= 24.47º enter cosine on the calculator 910.178279-3
= 910.178279-3 x 56’
= 50.96998362 (round off to 51’)
= 51’
Electronics: AC Waveforms Plan
24
AC Waveforms Exam
Given the following values, complete the table by determining the missing or incomplete
values for the rms, peak amplitudes, the peak-to-peak values, the average values and/or
the instantaneous values.
#
rms
13.
14.
peak
pk-to-pk
average
2.63uV
1.55uV @ 36º
________ V @ 165º
45V
15.
________ ____ @
70º
3.54pV
16.
17.
3.75V @ 70.7º
10.0kV
2.43kV @ 190º
18.
19.
20.
21.
instantaneous
82.1mV
72.3uV
40mV @ 18º
54.1uV @ 212º
905V
7.91V @ 179º
29.6mV
________ V @ 247º
Electronics: AC Waveforms Plan
25
AC Waveforms Exam Key
Given the following values, complete the table by determining the missing or incomplete
values for the rms, peak amplitudes, the peak-to-peak values, the average values and the
instantaneous values.
#
rms
peak
pk-to-pk
average
instantaneous
22.
1.86uV
2.63uV
5.26uV
1.68uV
1.55uV @ 36º
23.
45V
63V
126V
40V
16 V @ 165º
24.
1.25pV
1.77pV
3.54pV
1.13pV
-1.77pV @ 70º
25.
2.81V
3.97V
7.94V
2.53V
3.75V @ 70.7º
26.
10.0kV
14kV
28kV
8.92kV
2.43kV @ 190º
27.
91.2mV
129mV
258mV
82.1mV
40mV @ 18º
28.
72.3uV
102uV
204uV
65uV
54.1uV @ 212º
29.
320V
453V
905V
289V
7.91V @ 179º
30.
32.9mV
46.5mV
93mV
29.6mV
-42.8mV @ 247º
Electronics: AC Waveforms Plan