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Unit 1: Algebra 2/Trig Review. Unit 1 videos; Unit 1B: Polynomials and Rational Functions. Unit 1B video page; Unit 2: Exponential and Logarithmic Functions. Unit 2 videos; Unit 3: Triangle Trig and Unit Circle. Unit 3 videos; Unit 4: Graphing Trig Functions. Unit 4 videos; Unit 5: Trig Identities/ Solving Trig Equations. Unit 5 videos; Unit 6.
This is the homepage for the free book Trigonometry, by MichaelCorral (Schoolcraft College).
Algebra 2 Period 4. Welcome to Algebra 2. Algebra 2 is an extension of the Algebra 1 curriculum. Topics that were first introduced in Algebra 1 will be built upon and applied to problems that require higher order thinking skills. Quarter 3 Solutions for Enriched Trigonometry Algebra 5 2016-2017. Week 1 1-9-17 to 1-13-17. Hickman's Picture to return to Mr. Hickman's Main web page. The Map of Early Modern London comprises four distinct, interoperable projects. MoEML began in 1999 as a digital atlas of sixteenth- and seventeenth-century London based on the 1560s Agas woodcut map of the city. WEEK #1 - Proving Trigonometric Identities. NOTES - My notes in pdf format Video - Verifying Trig Identities Video - Verifying a trigonometric identity with multiple steps.
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Latest version (2020-09-25): Trigonometry.pdf
Code samples from the book: code_samples_trig.zip
Current changelog: changelog.txt
Note: The PDF was built using TeXLive 2011 and Ghostscript 9.27 underLinux (Fedora).
LaTeX source code:trigbook-1.2-src.tar.gz
The book is distributed under the terms of the GNU Free Documentation License,Version 1.3.
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You can buy a printed and bound paperback version of the bookwith grayscale graphics for $11 plus shipping at Lulu.comhere.
Book Description
This is a text on elementary trigonometry, designed for students who havecompleted courses in high-school algebra and geometry. Though designed forcollege students, it could also be used in high schools. The traditional topicsare covered, but a more geometrical approach is taken than usual. Also, somenumerical methods (e.g. the secant method for solving trigonometric equations)are discussed. A brief tutorial on using Gnuplot to graph trigonometricfunctions is included.
There are 495 exercises in the book, with answers and hints to selectedexercises.
Table of Contents
- Right Triangle Trigonometry
- Angles
- Trigonometric Functions of an Acute Angle
- Applications and Solving Right Triangles
- Trigonometric Functions of Any Angle
- Rotations and Reflections of Angles
- General Triangles
- The Law of Sines
- The Law of Cosines
- The Law of Tangents
- The Area of a Triangle
- Circumscribed and Inscribed Circles
- Identities
- Basic Trigonometric Identities
- Sum and Difference Formulas
- Double-Angle and Half-Angle Formulas
- Other Identities
- Radian Measure
- Radians and Degrees
- Arc Length
- Area of a Sector
- Circular Motion: Linear and Angular Speed
- Graphing and Inverse Functions
- Graphing the Trigonometric Functions
- Properties of Graphs of Trigonometric Functions
- Inverse Trigonometric Functions
- Additional Topics
- Solving Trigonometric Equations
- Numerical Methods in Trigonometry
- Complex Numbers
- Polar Coordinates
- Appendix A: Answers and Hints to Selected Exercises
- Appendix B: Graphing with Gnuplot
Miscellaneous
Go here to see a computer program (in theTcl programming language) for solving triangles.
Click here for aPDF file showing how to use trigonometric identities to find a simple closed-formexpression for sin 18° (and consequently cos 18°), which I prepared for my trigonometry class.Since expressions for sin 15° and cos 15° can be found byusing the difference formulas for sine and cosine (with 15° = 45° - 30°),the expressions for sin 18° and cos 18° can then be used again with thedifference formulas to get sin 3° and cos 3°, and hence the trigonometricfunctions of any integer multiple of 3°. Part of the purpose of doing thiswas to show how useful the product-to-sum formulas are; they do not seem to getthe attention they deserve in most textbooks these days—something I wanted torectify when I wrote my book (where I give a few examples of how thoseidentities are used in physics and electrical engineering).
News
(2021-01-05) Cleaned up the webpage to make it less hideous and more consistent with the revamped page forElementary Calculus.
(2015-07-19) After spending quite a bit of time trying to compile the book correctly with TeX Live 2014—andnot succeeding—I decided to give up and go back to TeX Live 2011. The reason for recompilingwas to make a few modifications and corrections.See the changelog for details.
(2012-06-14) Version 1.2 of Trigonometry is now out. The biggest change is the update of the license toversion 1.3 of the GNU Free Documentation License. This was long overdue, and I felt that therewere enough changes in the license to warrant a new version number for the book. I also updatedthe Gnuplot tutorial in Appendix B to reflect the current version (4.6.0) of the software, as wellas some stylistic changes. A few typos were corrected and some clarifications made (thanks to SethBraver at South Puget Sound Community College for bringing those to my attention).See the changelog for details.
(2011-11-04) Added a PDF file showing how to find an exact expression for sin 18°. Alsomade the Java applet for 2-dimensional rotation available again, but this timein a Zip file that can be downloaded, so that the user has to run it locally. Though thissite is now on a new hosting provider I am still worried about bandwidth usage,which is why I'm hesitant to have the live applet hosted on the site again.
(2011-08-13) In addition to being on a new hosting provider, a new update is out. This is primarilya “maintenance release” to clean up some oldcode so that it compiles under TeX Live 2010, which appears to handle fontsdifferently than previous versions; in particular the bbding and dingbat font packagesseem a bit buggy (the fonts get substituted, resulting in the wrongsymbols being displayed). So I removed the book's dependence on those packages andreplaced them with the pifont package, which gives a cleaner interface to thenecessary symbols. Similar to the Vector Calculus book, some spacing issues alsoarose, so those were fixed as well. Additionally, some of the EPS graphics wereimproved to have consistent fonts. Lastly, one typo was corrected. See thechangelog for details.
(2011-06-02) UnfortunatelyI had to remove the applets demonstrating Thales' Theorem and rotations in the xy-coordinateplane. I receiveda notice from my web hosting provider today that I was exceeding my bandwidth limit, and that myaccount would be suspended if I did not reduce the bandwidth within seven days. It's possiblethat the applets are the culprit; I'll find out soon enough.
(2010-12-01) The printed versionavailable at Lulu.com has (finally!) been updated with all the latestcorrections. I had been meaning to do this for a while and finally gotaround to it.
(2010-05-06) The book is now listed on theCLRN Free Digital Textbook Initiativepage. This means that the book met the CLRN (California Learning Resource Network)review criteria. Also created a changelog for documentingthe updates/corrections to the book.
(2010-04-18) Created aJava applet demonstrating rotations in two dimensions.
(2010-03-28) The book can now be bought as a printed, boundpaperback from Lulu.com.
(2010-03-12) Version 1.1 released. The only changes in materialare the addition of a section (6.4) on polar coordinates, and a short Sage example was addedto Section 6.2.
(2009-08-29) Initial version 1.0 is released.
| 'Dad says that anyone who can't use a slide rule is a cultural illiterate andshould not be allowed to vote. Mine is a beauty - a K&E 20-inch Log-log Duplex Decitrig' - Have Space Suit - WillTravel, 1958. by Robert A. Heinlein (1907-1988) | Downloadable Version of this Web Page. Use it on your classroom computers as a Seminar on your computer (4MB PDF) |
| This self-guided course gives numeric examples of the basic calculations that a sliderule can do. Just follow the step-by-step instructions and you will beamazed by the power and versatility of the venerable Slipstick. Click on any ofthe images below to get a large, unmarked, blowup of each slide rule as shown inthe problem. Images created from Emulators designed by Derek Ross.Initial examples based on Derek's Self Guided demo and expanded by Mike Konshak. | No Slide rule handy? Want to try a real Slide Rule? |
| Pickett Teaching Guide 17.7MB pdf
|
A Brief Timeline of the Slide Rule1614 - Invention of logarithms by John Napier, Baron of Merchiston, Scotland.1617 - Developments of logarithms 'to base 10' by Henry Briggs, Professor of Mathematics, Oxford University. 1620 - Interpretation of logarithmic scale form by Edmund Gunter, Professor of Astronomy, London. 1630 - Invention of the slide rule by the Reverend William Oughtred, London. 1657 - Development of the moving slide/fixed stock principle by Seth Partridge, Surveyor and Mathematician, England. 1775 - Development of the slide rule cursor by John Robertson of the Royal Academy. 1815 - Invention of the log log scale principle by P.M. Roget of France. 1850 - Amédée Mannheim, France, produced the modern arrangement of scales. 1886 - Dennert & Pape, Germany, introduce white celluloid as a material for inscription of scales. 1890 - William Cox of the United States patented the duplex slide rule. c1900 - Engine divided scales on celluloid increases precision of slide rules. 1976 - The final slide rule made by K&E donated to the Smithsonian Institute, Washington, DC, USA. Today, slide rules can be found on eBay, antique stores and estate auctions.Lost inventories of brand new slide rules turn up every year. | Was there Life |
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William Oughtred discovered the above characteristic in 1630, when he placed twologarithmic scales that were invented by his contemporary, Edmund Gunter,alongside each other. Thus the slide rule was born.
At this point it is best to just describe how to read the scales. On almost all slide rules, the black scales (A, B, C, D, K, etc.) increase from leftto right. The red scales, or inverse scales(CI, DI), increase from right to left. The pocket sized Pickett 600-ES will be used in most illustrations. The fullsized Pickett N3-T for others. By the way, the Pickett 600-T (white) was taken by the Apollo 11 NASA astronauts to the moon.
Except for 'folded', 'trig' or Log' scales, each scale beginswith 1. C and D scales are single logarithmic (1-10) scales. The A and B scales are double logarithmic(1-10-100) having two cycles of 1-10, the K scale being triple logarithmic (1-10-100-1000) having three cycles of1-10. The Primary divisions are whole numbers. The secondary divisions divide the Primary by 10, the Tertiary divisions divide the secondary by 5. Of course as you get to the end of each scale the divisions get so close together that the tertiary divisionsdisappear. The scales on each side of a slide rule are aligned so thatcalculations can be carried from one side to the other.
Its important to become familiar with not only the physical divisions as marked on the scales, but in becoming able toextrapolate* values when the hairline falls in the spacing betweendivisions. Positions of the slide and cursor shown in the examples will mention the label of the scale and the value on the scale, such as scale C at 1.5 will be referenced as C:1.5.
*In mathematics, extrapolation is the process of constructing new data points outside a discreteset of known data points. It is similar to the process of interpolation,which constructs new points between known points, but its results are often lessmeaningful, and are subject to greater uncertainty (Ref: en.wickipedia.org).| TOC |
1. Simple Multiplication (uses C and D scales)
Example: calculate 2.3 × 3.4 (Figure 1)- Slide the leftmost Index '1' on C over 2.3 on the D scale.
- Move the cursor to 3.4 on the C scale.
- The cursor is on the D scale just a bit over 7.8 or 7.82. This is the answer.
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2. 'Wrap-Around' Multiplication (uses C and D scales)
Example 2a: calculate 2.3 × 4.5 (Figure 2a)
- Move the left Index on C to 2.3 on the D scale.
- Try to move the cursor to 4.5 on the C scale. The cursor is blocked by the brace (also called strap or bridge).
- The target C:4.5 is off the D scale. The right Index must now be used.
Example 2b: calculate 2.3 × 4.5 (Figure 2b)
- Move the right Index on C (C:1) to over 2.3 on the D scale (D:2.3)
- Slide the cursor to 4.5 on the C scale (C:4.5)
- On the D scale, you'll see that the hairline is between divisions. Extrapolate the answer to 1.035.
- Do a rough approximation by rounding 2.3 to 2 and 4.5 to 5.
- We mentally calculate 2 × 5 = 10, so weadjust thedecimal place to get 10.35 or 10.4.
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3. Folded-Scale Multiplication (uses C, D, CF and DF scales)
Example 3: calculate 2.3 × 4.5 (Figure 3)- Slide the leftmost Index, '1', on C over the 2.3 on the D scale (D:2.3).
- We can't move the cursor to 4.5 on the C scale; it's out ofrange. We can use the folded scales to get this answer.
- Move the cursor to 4.5 on the CF scale (CF:4.5).
- The cursor is now at 1.04 on the DF scale (DF:1.04).
- We know the correct answer is near 2 × 5 = 10, so weadjust thedecimal place to get 10.4.
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4. Multiplication by π (uses D and DF scales)
Example 4: calculate 123 × π (Figure 4)
- Move the cursor to 1.23 on the D scale.
- The cursor is now at 3.86 on the DF scale.
- We know that the correct answer is near 100 × 3 = 300, soweadjust the decimal place to get 386.
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5. Simple Division (uses C and D scales)
Example 5: calculate 4.5 / 7.8 (Figure 5)- Move the cursor to 4.5 on the D scale.
- Slide 7.8 on the C scale to the cursor.
- Move the cursor to either the leftmost or rightmost '1' on the Cscale, whichever is in range. In this case, you would move it to therightmost '1'.
- The cursor is now at 5.76 on the D scale.
- We know that the correct answer is near 4/8 = 0.5, so we adjustthe decimal place to get 0.576.
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6. Reciprocal (uses C and CI scales)
Example 6: calculate the reciprocal of 7.8, or 1/7.8 (Figure 6)- Move the cursor to 7.8 on the CI scale. Note that the CI scaleincreases from right to left, as indicated by the '<' symbols beforethe numbers.
- The cursor is now at 1.28 on the C scale.
- We know that the correct answer is near 1/10 = 0.1, so we adjustthe decimal place to get 0.128.
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7. Sin(x) for angles between 5.7° and 90° (uses S and C scales)
Example 7: calculate sin(33°) (Figure 7)- Move the cursor to 33 on the S scale.
- The cursor is at 5.45 on the C scale.
- We know that the correct answer for a sin in this range isbetween 0.1 and 1, so we adjust the decimal place to get 0.545.
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8. Cos(x) for angles between 5.7° and 90° (uses S and C scales)
Example 8: calculate cos(33°). (Figure 8)- The cos scale shares the sin S scale. Instead of increasingfrom left to right like the sin scale, cos increases from right toleft. This is indicated on the slide rule by '<' characters whichremind you that the number in increasing 'backwards'.
- Move the cursor to <33 on the S scale.
- The cursor is now at 8.4 on the C scale.
- We know that the correct answer for a cos in this range isbetween 0.1 and 1, so we adjust the decimal place to get 0.84.
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9. Tan(x) for angles between between 5.7° and 45° (uses T and C scales)
Example 9: calculate tan(33°) (Figure 9).- Move the cursor to 33 on the T scale.
- The cursor is now at 6.5 on the C scale.
- We know that the correct answer for a tan in this range isbetween 0.1 and 1, so we adjust the decimal place to get 0.65.
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10. Tan(x) for angles between between 45° and 84° (uses backward T and CI scale)
Example 10a: calculate tan(63°) (Figures 10a and 10b)- Move the cursor to <63 on the T scale. Note that this rangeincreases from right to left, as indicated by the '<' before thenumbers.
- The cursor is now at <1.96 on the DI scale. Figure 10a
- If you flip to the other side the cursor is also at <1.96 on the CI scale. Figure 10b
- We know that the correct answer for a tan in this range isbetween 1 and 10, so we don't need to adjust the decimal place.
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11. Tan(x) for angles between between 45° and 84° (uses forward T and C scale)
Example 11: calculate tan(63°). (Figure 11)- Move the cursor to 63 on the forward T scale. This scaleincreases from left to right.
- The cursor is now at 1.96 on the C scale.
- We know that the correct answer for a tan in this range isbetween 1 and 10, so we don't need to adjust the decimal place.
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12. Sin(x) and tan(x) for angles between 0.6° and 5.7° (using the ST and C scales)
In this range, the sin and tan functions are very close in value, so asingle scale can be used to calculate both.Example: calculate sin(1.5°) (Figure 12)
- Move the cursor to 1.5 on the ST scale.
- The cursor is now at 2.62 on the C scale.
- We know that the correct answer for a sin in this range isbetween 0.01 and 0.1, so we adjust the decimal place to get 0.0262.
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13. Sin(x) and tan(x) for other small angles (using C and D scales)
For small angles, the sin or tan function can be approximated closelyby the equation:sin(x) = tan(x) = x / (180/π) = x / 57.3.
Knowing this, the calculation becomes a simple division. This techniquecan also be used on rules without an ST scale.
Example: calculate sin(0.3°) (Figure 13)
- Move the cursor to 3 on the D scale.
- Slide 5.73 on the C scale to the cursor. Most rules have a ticklabeled 'R' at this point.
- Move the cursor to either the leftmost or rightmost '1' on the Cscale, whichever is in range.
- The cursor is now at 5.24 on the D scale.
- We know that the correct answer is near 0.3 / 60 = 0.005, so weadjust the decimal place to get 0.00524.
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14. Square (uses C and B scales)
Example 14: calculate 4.7 2Trigonometry Videosmr. Mackenzie's Web Page Sign In
- Move the cursor to 4.7 on the C scale.
- The cursor is now at 2.2 on the B scale.
- We know that the correct answer is near 52 = 25, so weadjust the decimal place to get 22.
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15. Square Root (uses C and B scales)
Example 15a: calculate √4500 (Figure 15)- You will notice that the B scale has two similar halves. Thefirst step is to decide which half to use to find a square root.
- The left half is used to find the square root of numbers with oddnumbers of digits or leading zeros after the decimal point. The righthalf is used for numbers with even numbers of digits or leading zeros.Since 4500 has an even number of digits, then we'll use the right halfof the scale.
- Move the cursor to 4.5 on the right half of the B scale.
- The cursor is now at 6.7 on the C scale.
- We know that 702 = 4900, which is in the ballpark of4500. Therefore we adjust the decimal point to get a result of 67.
Example 15b: calculate √450 (Figure 15b)
- Try it a again with a three digit number.
- Move the cursor to 4.5 on the left half of the B scale.
- The cursor is now at 2.12 on the C scale.
- We know that 202 = 400, which is in the ballpark of450. Therefore we adjust the decimal point to get a result of 21.2
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16. Cube (uses D and K scales)
Example 16: calculate 4.73 (Figure 16)- Move the cursor to 4.7 on the D scale.
- The cursor is now at 1.04 on the K scale.
- We know that the correct answer is near 5 x 5 x 5, which, tofurther approximate, is near 5 x 5 x 4 = 5 x 20 = 100. Therefore weadjust the decimal point to get a result of 104.
Trigonometry Videosmr. Mackenzie's Web Page Template
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17. Cube Root (uses D and K scales)
Example 17a: calculate 3√4500 (Figure 17 - left)- You will notice that the K scale has three similar thirds. Thefirst step is to determine which third to use to find the cube root.
- The first third is used to find the cube root is numbers with onedigit. You can cycle through the thirds, increasing the number ofdigits by onefor each third, to find which part to use. Just like having a line of people and you count off into 3's.
- For the value of 4500, which has 4 digits, we cycle through thethirds and find that we would use the first third.
- Move the cursor to 4.5 on the first third of the K scale.
- The cursor is now at 1.65 on the D scale.
- We can take a guess that the correct answer is around 10. Thecube of 10 is 1000 and the cube of 20 is 8000. Thus we know that thecorrect answer is between 10 and 20, therefore we can move the decimalplace and get the correct result of 16.5.
Example 17b: calculate 3√450000 (Figure 17 - right)
- For the value of 450000, which has 6 digits, we cycle through thethirds and find that we would use the third third.
- Move the cursor to 4.5 on the third third of the K scale.
- The cursor is now at 7.68 on the D scale.
- We can take a guess that the correct answer is around 10. Thecube of 10 is 1000 and the cube of 100 is 1000000. Thus we know that thecorrect answer is between 10 and 100, therefore we can move the decimalplace and get the correct result of 76.8.
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Since there are many slight variations of log-log scales on differentslide rules, I'll refer only to the scales found on the Pickett N3,Pickett N600 and Pickett N803 slide rules (among others). If you wantto view a virtual N3, click here,if you want a virtual N600, click here(opens in a new window.)
Another interesting aspect of LL scales is that the decimal pointis 'placed.' That is, you don't have to figure out afterwards where thedecimal point belongs in your result. The disadvantage to this is thatLL scales are limited in the numbers they can calculate. Typically, thehighest result you can get is about 20,000, and the lowest is 1/20,000or 0.00005. One exception to this is the Picket N4 (virtual here),which goes up to 1010.
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18. Raising a Number to Powers of 10 (N>1)
To raise a number to the power of 10, simply move the cursor to thenumber and look at the next highest LL scale. (These examples are fornumbers greater than 1. )Example 18a: calculate 1.35 10 (uses LL2andLL3 scales) (Figure 18a)
- Move the cursor to 1.35 on the LL2 scale.
- The cursor is at 20.1 on the LL3 scale. This is the correctanswer.
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Example 18b: calculate 1.04 100 (uses LL1andLL3 scales) (Figure 18b)
- Set the cursor over 1.04 on the LL1 scale.
- Flip the SR over
(on some advanced SR's the LL0, LL1, LL2 and LL3 will be on the same side) - The cursor is at 50.5 on the LL3 scale.
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Example 18c: calculate 1.002 1000 (uses LL0 and LL3scales) (Figure 18c)
- Set the cursor over 1.002 on the LL0 scale.
- The cursor is on 7.4 on the LL3 scale. This is the correct answer.
Example 18d: calculate sequential powers of ten of 1.002 (uses LL0, LL1, LL2 andLL3scales) (use the above Figure 18c)
- Set the cursor to 1.002 on the LL0 scale. This the same as 1.002 1
- On LL1, the cursor is at 1.002 10 , or 1.020
- On LL2, the cursor is at 1.002 100 , or 1.22.
- On LL3, the cursor is at 1.002 1000 , or 7.37.
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19. Raising a Number to Powers of 10 (N<1)
The reciprocals of the LL scales are the -LL scales. They work the sameway, but you have to make sure that you look for the answer on a -LLscale. They are RED scales so they increase invalue from right to left.Example 19: calculate 0.75 10 (uses -LL2and -LL3 scales)
- Move the cursor to <0.75 on the -LL2 scale.
- The cursor is at <0.056 on the -LL3 scale. This is the correctanswer.
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20. Raising a Number to Powers of -10 (N<1), N-10 or 1/N10
The reciprocals of the LL scales are the -LL scales. They work the sameway, but you have to make sure that you look for the answer on a -LLscale.Example 20: calculate 1.175 -10 (usesLL2and -LL3 scales)
- Set the cursor hairline to 1.175 on the LL2 scale.
- The hairline is over 5 on the LL3 scale. This is 1.175 10
- The cursor is also over .2 on the -LL3 scale. This is the reciprocal of 1.175 -10 or 1/1.175 10 and is the correctanswer.
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21. Finding the 10th Root (N>1)
As you've seen in the previous examples, to raise a number to the 10thpower, you simply look at the adjacent number on the next highest LLscale. To find a tenth root, you look at the adjacent number on thenext lowest LL scale. Remember also that finding the tenth root is thesame as raising a number to the power of 0.1.Example 21: calculate 10√5, or 5 0.1 (uses LL2 and LL3 scales) (Figure 21)
- Set the cursor over 5 on the LL3 scale.
- The cursor is now at 1.175 on the LL2 scale. This is the correctanswer.
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22. Finding the 10th Root (N<1)
Example: calculate 100√0.15, or 0.15 0.01 (uses -LL3 and -LL1 scale) (Figure 22)- Set the cursor to 0.15 on the -LL3 scale.
- Flip the SR to the other side to read the LL1 scale.
- The cursor reads 0.9812 on the -LL1 scale. This is thecorrect answer.
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23. Arbitrary Powers (Staying on Same LL Scale)
Occasionally, depending on the numbers, it is possible to calculate apower without switching scales.Example: calculate 9.1 2.3 (uses LL3 scale) (Figure 23a)
- Set the cursor hairline to 9.1 on the LL3 scale.
- Slide the leftmost Index '1' on C to the hairline.
- Move the hairline over 2.3 on the C scale.
- The cursor is now at about 161 on the LL3. This is very close tothe correct answer of 160.6. One of the problems with LL scales is thattheir accuracy diminishes as the numbers increase in value.
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Example: calculate 230 0.45 (usesLL3 and C scale) (Figure 23b)
- Set the cursor to 230 on the LL3 scale.
- Since we're raising to a power that's less than 1, we have to goleft on the LL scale.
- Slide the rightmost Index '1' on the C scale to the cursor hairline.
- Move the cursor to 4.5 on the C scale.
- The cursor is now at 11.6 on the LL3 scale. This is close to thecorrect answer of 11.56.
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- Set the cursor to 0.78 on the -LL2 scale.
- Slide the leftmost Index '1' on the C scale to the cursor hairline.
- Move the cursor to 3.4 on the C scale.
- The cursor is now at 0.43 on the -LL2 scale. This is the correctanswer.
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Example: calculate 0.78 0.45 (uses -LL2 and C scale)(Figure 23d)
- Set the cursor to 0.78 on the -LL2 scale.
- Since we're raising to a power that's less than 1, we have to goleft on the LL scale.
- Slide the rightmost Index '1' on the C scale to the cursor hairline.
- Move the cursor to 4.5 on the C scale.
- The cursor is now at 0.894 on the -LL2 scale. This is the correctanswer.
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24. Arbitrary Powers (Switching LL Scales)
One of the rules of exponents is that (A B ) Cis equal to A B x C. We can use this fact, along with ourknowledge of powers of ten, to calculate arbitrary powers.Example: calculate 1.9 2.5 (uses LL2, C and LL3 scales) (Figure 24a)
- If we try to calculate this the easy way, the power 2.5 is out ofrange for the scale.
- We can reinterpret the problem as:
Calculate (1.9 0.25 ) 10
Because 0.25 x 10 is 2.5. - Set the cursor hairline to 1.9 on the LL2 scale.
- Slide the rightmost Index '1' on the C scale to the hairline.
- Shift the cursor hairline to 2.5 on the C scale.
- The hairline is now at 1.9 0.25 on the LL2 scale. Sincewe want to also raise this to the power of 10, we look 'one scalehigher' at the LL3 scale.
- The cursor is at 4.97 on the LL3 scale. This is the correctanswer.
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- Like the previous example, if we try to calculate this the easyway, the power 0.34 is out of range for the scale.
- We can reinterpret the problem as:
Calculate (12 3.4 ) 0.1
Because 3.4 x 0.1 is 0.34. - Move the cursor to 12 on the LL3 scale.
- Slide the leftmost '1' on the C scale to the cursor.
- Move the cursor to 3.4 on the C scale.
- The cursor is now at 12 3.4 on the LL3 scale, whichis about 5000 (which is not the number we're looking for). Since wealso want to raise this to the power of 0.1, we look at the LL2 scale.
- The cursor is now at 2.33 on the LL2 scale. This is the correctanswer.
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- We can reinterpret this problem as:
Calculate (0.99 5.6 ) 100
Because 5.6 x 100 = 560 - Set the cursor Hairline to <0.99 on the -LL1 scale.
- Shift the leftmost Index '1' on the C scale to the cursor hairline.
- Move the cursor to 5.6 on the C scale.
- The cursor is now at <0.99 5.6 = <.9452 on the -LL1 scale.
- Since we also want to raise this to the power of 100, we look 'twoscales higher', or the -LL3 scale.
- The cursor is now at <0.0036 on the -LL3 scale. This is thecorrect answer.
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25. Log-Log Approximations
In general, LL scales don't handle numbers extremely close to 1, suchas 1.001 or 0.999. This is not a problem because there is an accurateapproximation for numbers in this range. In general, if you have a verysmall number 'd', then:(1 + d) p = 1 + d p
Example: calculate 1.00012 34 (uses C and D scales) (Figure25a)
- In this case, if we use the approximation (1 + d) p =1 + d p, then:
d = 0.00012, and
p = 34 - We must calculate 0.00012 * 34.
- Set the leftmost Index '1' on the C scale to 1.2 on the D scale..
- Set the cursor hairline on 3.4 on the C scale.
- The cursor is now at 4.08 on the D scale.
- We know that the correct answer would be near 0.0001 * 30, or0.003. Therefore we adjust the decimal point to get a value of 0.00408.
- Add 1 to 0.00408. The result is 1.00408, which is very close tothe correct answer of 1.004088.
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Trigonometry Videosmr. Mackenzie's Web Page Pdf
- Like before, we'll use the approximation (1 + d) p =1 + d p. In this case:
d = (0.99943 - 1) = -0.00057, and
p = 21 - We must calculate -0.00057 * 21.
- Slide the rightmost Index '1' on the C scale to 5.7 on the D scale.
- Move the cursor hairline over 2.1 on the C scale.
- The hairline is at 1.195 on the D scale.
- We know that the correct answer would be near -0.0006 * 20, or-0.0120. Therefore we adjust the decimal point to get a value of-0.01195.
- Subtract 0.01195 from 1. The result is (1 - 0.01195) = 0.98803.This is very close to the correct answer of 0.98809.
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If you are an educator or home schooler wishing to give your students a hands-on experience and instruction withactual slide rules, the museum is able to supply quantities of up to 25 matching slide rules for temporary use, free of charge, to many countries, courtesy of several collectors and members of the Oughtred Society, Dutch Kring, and German RST. Find out more about it here.
You can download additional instructions from theISRM museum libraryor visit one of the following sites.
In the beginning, at the time of the great flood, Noah went thru his ark after it landed, and found two small snakes huddled in a corner. Noah looked at these poor specimens - and said 'I told you to go forth and multiply - why haven't you?' The poor snakes looked up at Noah and replied 'We can't because we are adders.....' Noah looked a bit perplexed, and then proceeded to tear bits of planking from his ark. He went on to build a beautiful wooden platform. He gathered up the snakes and placed them on the platform, and joyfully told the snakes - 'Now go forth and multiply, because even adders can multiply on a log table' D. Scott MacKenzie, PhD Modifications, layout and graphics Copyright © 2006 by Mike Konshak |