Precision Sheet Metal Workshop Mathematics

Transcription

1 Precision Sheet Metal Workshop Mathematics The Order of operations Highest priority: Brackets [ ] Parentheses ( ) Secondary priority: Squares x² Square-roots Third highest priority: Multiplication X Division / Lowest priority: Addition + Subtraction- 1

2 Algebra - simple questions pt.1 1) = 2 2) = 2 Let s look at some simple examples! 3) * 3 = 8 4) 4 / = 4 Algebra - simple questions pt.2 5) 2 * (5 + 5) = 20 6) (5-3) * 2 = 4 It looks hard.. but it s not! 7) [(5 + 5) * 2 ] + (2 * 4) = 28 8) [5 * (2 + 2)] * [(2 + 2) - (2-1)] = 60 2

3 Number Lines Cartesian Coordinates It s a 3-dimensional Number Line 3

4 In math and in sheet metal it s all about understanding the terminology, for example what s a Coefficient Coefficients 4

5 Pythagorean Theorem of right triangles claims that: The sum of the square root of any two sides of a right triangle is equal to the sum of the remaining side. Pythagorean theorem A 2 + B 2 = C 2 5

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7 Constants are no more than a Fixed Value! 7

8 Can great thinkers get it wrong? Did the Scarecrow get it? Lets find out. After all he has a brain and a diploma! Courtesy of MGM 81 is the answer to this math limerick (5 x 11) = A dozen, a gross, and a score plus three times the square root of four. Divided by seven + five times eleven is Nine squared and not a bit more 8

9 Acute triangle all angles are acute. Triangle types Equiangular triangle all angles are acute and congruent. Obtuse triangle one angle is obtuse. Right triangle one angle is a right angle Trig tables It s an older but valid way to solve trig problems! 9

10 We need to solve triangles on the prints to finding missing dimensions, layout notches and solve bend functions Bend Functions like Outside Setbacks OSSB 10

11 The Formulas We can find the Angles or we can find the sides! Convention: Labels and values A, B, and C (uppercase) are angles A is always equal to 90 degrees a, b, and c (lowercase) are the sides AKA a the Hypotenuse b the Adjacent side c the Opposite side 11

12 Convention: Right Angle Truths The Angles A = 90 degrees B = Steepest angle C = Shallowest angle Angles always total 180 degrees!!! When angles B and C = 45 degrees, sides b and c are equal! The Sides a = Longest side b = Middle length c = Shortest side Information that s required? Two angles and a side or two sides and an angle? 12

13 Calculator keys and functions: Trig function keys Sine, Co-sine, Tangent Memory keys RM = Recall memory X-M = clear memory M+ = add to memory Number key pad Parentheses Degree, Radians, Grads For our use, the calculator needs to be in degree mode Math functions Addition, subtraction, multiplication and division You ll need this knowledge! 13

14 Calculator keys and functions: Inverse functions Yellow shift key (INV) Invokes functions written above each key. e.g.: sine to angle to sine Some functions we don t use. Secant s Co-Secant s Co-Tangent s 14

15 The calculator doesn t have keys that perform those functions. Top row: M, Sine, Cosine, Tangent, Cotangent, Secant, Cosecant, M Degrees 0-45 Values for the functions Degrees Bottom row (reversed of top): M, Cosine, Sine, Cotangent, Tangent, Cosecant, Secant, M 15

16 A quick look at what the different functions mean! Mode of operation Calculator must be in degree mode. To change modes push the DEG key. 16

17 Notes on the upcoming problems: 1) Your answer may not match ours because of how your calculator carries out the decimal point. 2) Always use the values that you begin with to avoid a compounding error. Here use the data given in BOLD as the values that you work with. Trigonometry an example from the tables: A = 90 B = MISSING C = a = b = c =.500 To find B search chart for formula i.e.: b/a = Sine of B / =

18 Trig table with sine value Read up the sine column from bottom. Sine of Angle B = Angle of 74 degrees 45 minutes The Answer is A = 90 B = C = a = b = c =.500 To find B search chart for formula i.e.: b/a = Sine of B / = = 74 degrees 45 minutes 18

19 #1 A = 90 B = 45 C = 45 a =.353 b =.250 c =.250 Angles column 2 row 7 B = A - C Sides column 2 row 10 b = c / Tan C or b = c Sides column 2 row 2 a = c / sine C #2 A = 90 B = 87 C = 3 a =.400 b =.399 c =.021 Angles column 2 row 1 C = A - B Sides column 1 row 7 b = a x sine B Sides column 1 row 12 c = a x Cos B 19

20 #3 A = 90 B = C = a =.632 b =.600 c =.200 Angles column 1 row 9 b / c= Tan B Angles column 2 row 1 C = A - B Sides column 1 row 1 ( b 2 + c 2 ) #4 A = 90 B = C = a = b = c =.500 Angles column 1 row 8 c / a = Cos B Angles column 2 row 1 C = A - B Sides column 1 row 6 ( a 2 - c 2 ) 20

21 Our contestant says his protractor can t measure degrees! It s easy, you just convert decimal degrees to degrees, minutes and seconds, Degree to Decimal and back: Charts Decimal to minutes Decimal to seconds 21

22 To convert (21 minutes, 36 seconds) to decimal degrees 21 = = = To convert decimal degrees back to (21 minutes, 36 seconds) = The DEG / DMS key DEG key changes decimals to degrees, minutes and seconds DMS (inverse) key changes degrees, minutes, and seconds to a decimal 22

23 Finding a circular flat blank will require some math Circumference rulers Are not a precision method! A 3-inch diameter Equals a 9-3/8 inch circumference 23

24 Circumference = Pi x Diameter A precision method!

25 π Is a numeric value used to calculate the circumference of a circle Pi = Diameter x π = flat length Follow along here x = x =

26 Subtract one Material Thickness from the outside Diameter and use that Value to calculate the circumference What is a Radian? It s just a different way to divide a circle up. 26

27 Degrees of angle (included) x ( / 180) 45 X ( /180) = radians 45 X =.785 radians The interesting thing here is the mirror image of PI.E 27

28 Metric Conversion Either or 25.4 can be used to convert! 25 millimeters x =.984 inches / =.984 inches 25 millimeters 25 millimeters / 25.4 =.984 inches.984 inches x 25.4 = 25 millimeters 28

29 Power of a Number, Exponents 29

30 Calculating percentages ¾ = x 100 = 75 or 75% It s just messing around with percentages Great thinkers do get it wrong sometimes! Courtesy of MGM 30

31 The Scarecrow got it wrong! Pythagoras' Theorem of right triangles claims that: The sum of the square root of any two sides of a right triangle is equal to the sum of the remaining side. What Scarecrow states it is an isosceles triangle Have trouble with this section? It s okay, you re in good company! Okay one last math question, mathematically speaking, how many Cat are in this box? (cat + cat 3 ) 3 (1 + 1) 3 (2) 3 2 x 2 x 2 Answer: 8 cat 31

32 Conclusion The tools are easy to use... The mathematics are basic... And the basis for everything yet to come! 32