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 UNT in partnership with TEA. Copyright ©. All rights reserved. 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) – Reading/comprehension • 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 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 • Adjacent • Sine • Cosine • Tangent • • PowerPoint slides 26-30 Student Guide pages 6-7 8. Use Ratios and Formulas in Right-Side Trigonometric Functions • Opposite • Hypotenuse • Adjacent • 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 7 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 8 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 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 = adjacent 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 adjacent Electronics: AC Waveforms Plan UNT in partnership with TEA. Copyright ©. All rights reserved. 12 Right Triangle Hypotenuse Opposite Adjacent Formulas Opposite Sine Adjacent Hypotenuse Cosine Hypotenuse Opposite Tangent Adjacent Electronics: AC Waveforms Plan UNT in partnership with TEA. Copyright ©. All rights reserved. 13 Triangle #1 Triangle #2 Hypotenuse? Hypotenuse? 8' Opposite 10' Adjacent 5.3 Rods Opposite 6.8 Rods Adjacent Hypotenuse: Hypotenuse: Triangle #3 Triangle #4 125 miles Hypotenuse? 56' Hypotenuse? Opposite? 85 miles Adjacent Opposite: 23.2' Opposite Adjacent? Adjacent: Electronics: AC Waveforms Plan UNT in partnership with TEA. Copyright ©. All rights reserved. 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 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 = adjacent 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 adjacent Electronics: AC Waveforms Plan UNT in partnership with TEA. Copyright ©. All rights reserved. 20 Right Triangle Hypotenuse Opposite Adjacent Formulas Opposite Sine Adjacent Hypotenuse Cosine Hypotenuse Opposite Tangent Adjacent Electronics: AC Waveforms Plan UNT in partnership with TEA. Copyright ©. All rights reserved. 21 Triangle #1 Triangle #2 Hypotenuse? Hypotenuse? 8' Opposite 10' Adjacent Hypotenuse: 5.3 Rods Opposite 6.8 Rods Adjacent 12.8' Triangle #3 Hypotenuse: 8.62 Rods Triangle #4 125 miles Hypotenuse? 56' Hypotenuse? Opposite? 85 miles Adjacent Opposite: 91.7 miles 23.2' Opposite Adjacent? Adjacent: 51' NOTE: See next page for detail answers. Electronics: AC Waveforms Plan UNT in partnership with TEA. Copyright ©. All rights reserved. 22 Triangle #1 Hypotenuse? Adjacent: 10’ Opposite: 8’ 1st step Find the degree angle. Tangent = Opposite Adjacent =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. Hypotenuse = Adjacent Cosine = ___10’ __ take 38.66° enter cosine on your calculator your answer is 0.780866719 .780866719 = 12.8’ Triangle #2 Hypotenuse? Adjacent: 6.8 rods Opposite: 5.3 rods 1st step Find the degree angle. Tanget = Opposite Adjacent = 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 23 Triangle #3 Hypotenuse: 125 miles Adjacent: 8.5 miles Opposite:? 1st step Find the degree angle. Cosine = _Adjacent__ 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’ Adjacent: ? 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 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 UNT in partnership with TEA. Copyright ©. All rights reserved. 26