Trigonometry, Measurement, and Binary

3-24 h r diameter bore piston d piston d Piston at top center Piston at bottom center H=stroke cylinder concept from that of volume. For example, surface area is the amount of sheet metal needed to build a rectangular fuel tank while volume is the amount of fuel that the tank can contain. Rectangular Solid The formula for the surface area of a rectangular solid [Figure 3-24] is given as: Surface area = 2 × [(width × length) + (width × height) + (length × height)] = 2 × [(w × l) + (w × h) + (l × h)] Cube The formula for the surface area of a cube [Figure 3-25] is given as: Surface area = 6 × (side × side) = 6 × s2 Example: What is the surface area of a cube with a side measure of 8 inches?

Surface area = 6 × (side × side) = 6 × S2 = 6 × 82 = 6 × 64 = 384 square inches Cylinder The formula for the surface area of a cylinder [Figure 3-26] is given as: Surface area = 2 × π × radius2 + π × diameter × height = 2 × π × r2 + π × d × h Sphere The formula for the surface area of a sphere [Figure 3-28] is given as: Surface area = 4 × π × radius2 = 4 × π × r2 Cone The formula for the surface area of a right circular cone [Figure 3-29] is given as: Surface area = π × radius × [radius + √(radius2 + height2)] = π × r × [r + √(r2 + h2)] volume and surface area of three-dimensional solids.

Trigonometric Functions

Trigonometry is the study of the relationship between the angles and sides of a triangle. The word trigonometry comes from the Greek trigonon, which means three angles, and metro, which means measure. Right Triangle, Sides, and Angles In Figure 3-31 , notice that each angle is labeled with a capital letter. Across from each angle is a corresponding side, each labeled with a lower case letter. This triangle is a right triangle because angle C is a 90° angle. Side “a” is 3-25 h s r opposite from angle A and is sometimes referred to as the “opposite side.” Side “b” is next to, or adjacent to, angle A and is therefore referred to as the “adjacent side.” Side “c” is always across from the right angle and is referred to as the “hypotenuse.” Sine, Cosine, and Tangent The three primary trigonometric functions and their abbreviations are: sine (sin), cosine (cos), and tangent (tan). These three functions can be found on most scientific calculators. The three trigonometric functions are actually ratios comparing two of the sides of the triangle as follows: Sine (sin) of angle A = opposite side (side a) hypotenuse (side c) Cosine (cos) of angle A = adjacent side (side b) hypotenuse (side c) Tangent (tan) of angle A = opposite side (side a) adjacent side (side b) Example: Find the sine of a 30° angle.

Calculator Method: Using a calculator, select the “sin” feature, enter the number 30, and press “enter.” The calculator should display the answer as 0.5. This means that when angle A equals 30°, then the ratio of the opposite side (a) to the hypotenuse (c) equals 0.5 to 1, so the hypotenuse is twice as long as the opposite side for a 30° angle. Therefore, sin 30° = 0.5. Trigonometry Table Method: When using a trigonometry table, find 30° in the first column. Next, find the value for sin 30° under the second column marked “sine” or “sin.” The value for sin 30° should be 0.5. Pythagorean Theorem The Pythagorean Theorem is named after the ancient Greek mathematician, Pythagoras (~500 B.C.). This theorem is used to find the third side of any right triangle when two sides are known. The Pythagorean Theorem states that a 2 + b 2 = c 2.

[Figure 3-32] Where “c” = the hypotenuse of a right triangle, “a” is one side of the triangle and “b” is the other side of the triangle. Example: What is the length of the longest side of a right triangle, given the other sides are 7 inches and 9 inches? The longest side of a right triangle is always side “c,” the hypotenuse. Use the Pythagorean Theorem to solve for the length of side “c” as follows: a 2 + b2 = c2 7 2 + 92 = c2 49 + 81 = c2 130 = c2 c = 130 = 11.4 inches Therefore, side “c” = 11.4 inches. Example: The cargo door opening in a military airplane is a rectangle that is 5 1⁄2 feet tall by 7 feet wide. A section of square steel plate that is 8 feet wide by 8 feet tall by 1 inch thick must fit inside the airplane. Can the square section of steel plate fit through the cargo door? It is obvious that the square steel plate will not fit horizontally through the cargo door. The steel plate is 8 feet wide and the cargo door is only 7 feet wide. However, if the steel plate is tilted diagonally, will it fit through the cargo door opening?

The diagonal distance across the cargo door opening can be calculated using the Pythagorean Theorem where “a” is the cargo door width, “b” is the cargo door height, and “c” is the diagonal distance across the cargo door opening. a 2 + b2 = c2 3-26 Solid Volume Surface Area Figure 3-24 3-25 3-26 3-28 3-29 l × w × h s3 π × r2 × h ⁴⁄3 × π × r3 ¹⁄3 × π × r2 × h 2 × [(w × l) + (w × h) + (l × h)] 6 × s2 2 × π × r2 + π × d × h 4 × π × r2 π × r × [r + √(r2 + h2)] Rectangle Solid Cube Cylinder Sphere Cone c b a C B A c b a a2 + b2 = c2 (7 ft)2 + (5.5 ft)2 = c2 49 + 30.25 = c2 79.25 = c2 c = 8.9 ft The diagonal distance across the cargo door opening is 8.9 feet, so the 8-foot wide square steel plate fits diagonally through the cargo door opening and into the airplane.

Measurement Systems

Conventional (U.S. or English) System Our conventional (U.S. or English) system of measurement is part of our cultural heritage from the days when the thirteen colonies were under British rule. It started as a collection of Anglo-Saxon, Roman, and Norman-French weights and measures. For example, the inch represents the width of the thumb and the foot is from the length of the human foot. Tradition holds that King Henry I decreed that the yard should be the distance from the tip of his nose to the end of his thumb. Since medieval times, commissions appointed by various English monarchs have reduced the chaos of measurement by setting specific standards for some of the most important units. Some of the conventional units of measure are: inches, feet, yards, miles, ounces, pints, gallons, and pounds. Because the conventional system was not set up systematically, it contains a random collection of conversions. For example, 1 mile = 5,280 feet and 1 foot = 12 inches.

Metric System The metric system, also known as the International System of Units (SI), is the dominant language of measurement used today. Its standardization and decimal features make it well- suited for engineering and aviation work. The metric system was first envisioned by Gabriel Mouton, Vicar of St. Paul’s Church in Lyons, France. The meter is the unit of length in the metric system, and it is equal to one ten-millionth of the distance from the equator to the North Pole. The liter is the unit of volume and is equal to one cubic decimeter. The gram is the unit of mass and is equal to one cubic centimeter of water.

All of the metric units follow a consistent naming scheme, which consists of attaching a prefix to the unit. For example, since kilo stands for 1,000, one kilometer equals 1,000 meters. Centi is the prefix for one hundredth, so one meter equals one hundred centimeters. Milli is the prefix for one thousandths and one gram equals one thousand milligrams. Refer to Measurement Systems & Conversions The United States primarily uses the conventional (U.S. or English) system, although it is slowly integrating the metric system (SI). A recommendation to transition to the metric system within ten years was initiated in the 1970s.

However, this movement lost momentum, and the United States continues to use both measurement systems. Therefore, information to convert between the conventional (U.S. or English) system and the metric (SI) system has been included in Figure 3-23 , Applied Mathematics Formula Sheet. Examples of its use are as follows: To convert inches to millimeters, multiply the number of inches by 25.4. Example: 20 inches = 20 × 25.4 = 508 mm To convert ounces to grams, multiply the number of ounces by 28.35. Example: 12 ounces = 12 × 28.35 = 340.2 grams 3-27 Prefix Multiplier (Exponential) Multiplier (Numerical) MeaningSymbol exa (1018) peta (1015) tera (1012) giga (109) mega (106) kilo (103) hecto (102) deca (101) unit deci (10−1) centi (10−2) milli (10−3) micro (10−6) nano (10−9) pico (10−12) femto (10−15) atto (10−18) quintillion quadrillion trillion billion million thousand hundred ten tenth hundredth thousandth millionth billionth trillionth quadrillionth quintillionth 1,000,000,000,000,000,000 1,000,000,000,000,000 1,000,000,000,000 1,000,000,000 1,000,000 1,000 100 10 0.1 0.01 0.001 0.000,001 0.000,000,001 0.000,000,000,001 0.000,000,000,000,001 0.000,000,000,000,000,001 E P T G M k h da 1 d c m µ n p f a Less Than 1 Greater Than 1

The Binary Number System

The binary number system has only two digits: 0 and 1. The prefix in the word “binary” is a Latin root for the word “two” and its use was first published in the late 1700s. The use of the binary number system is based on the fact that switches or valves have two states: open or closed (on/off). Currently, one of the primary uses of the binary number system is in computer applications. Information is stored as a series of 0s and 1s, forming strings of binary numbers. An early electronic computer, ENIAC ( Electronic Numerical Integrator and Calculator), was built in 1946 at the University of Pennsylvania and contained 17,000 vacuum tubes, along with 70,000 resistors, 10,000 capacitors, 1,500 relays, 6,000 manual switches and 5 million soldered joints. Computers obviously have changed a great deal since then, but are still based on the same binary number system. The binary number system is also useful when working with digital electronics because the two basic conditions of electricity, on and off, can be represented by the two digits of the binary number system. When the system is on, it is represented by the digit 1, and when it is off, it is represented by the digit 0.

Place Values The binary number system is a Base-2 system. That is, the place values in the binary number system are based on powers of 2. An 8-bit binary number system is shown in Converting Binary Numbers to Decimal Numbers To convert a binary number to a decimal number, add up the place values that have a 1 (place values that have a zero do not contribute to the decimal number conversion). Example: Convert the binary number 10110011 to a decimal number. Using the place value chart in Figure 3-35, add up the place values of the ‘1s’ in the binary number (ignore the place values with a zero in the binary number).

The binary number 10110011 = 128 + 0 + 32 + 16 + 0 + 0 + 2 + 1 = 179 in the decimal number system Converting Decimal Numbers to Binary Numbers To convert a decimal number to a binary number, the place values in the binary system are used to create a sum of numbers that equal the value of the decimal number being converted. Start with the largest binary place value and subtract from the decimal number. Continue this process until all of the binary digits are determined. Example: Convert the decimal number 233 to a binary number. Start by subtracting 128 (the largest place value from the 8-bit binary number) from 233.

233 – 128 = 105 A “1” is placed in the first binary digit space: 1XXXXXXX. Continue the process of subtracting the binary number place values: 105 – 64 = 41 A “1” is placed in the second binary digit space: 11XXXXXX. 41 – 32 = 9 A “1” is placed in the third binary digit space: 111XXXXX. 3-28 Place Value 27 or 128 26 or 64 25 or 32 24 or 16 23 or 8 22 or 4 21 or 2 20 or 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 1 10011001 shown as 00101011 shown as =153 =43 Place Value 27 or 128 26 or 64 25 or 32 24 or 16 23 or 8 22 or 4 21 or 2 20 or 1 1 0 1 1 0 0 1 1 128 + 0 + 32 + 16 + 0 + 0 + 2 + 1 10110011 shown as =179 Place Value 27 or 128 26 or 64 25 or 32 24 or 16 23 or 8 22 or 4 21 or 2 20 or 1 0 0 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 35 shown as 124 shown as 96 shown as 255 shown as 233 shown as Since 9 is less than 16 (the next binary place value), a “0” is placed in the fourth binary digit space, 1110XXXX.

9 – 8 = 1 A “1” is placed in the fifth binary digit space: 11101XXX Since 1 is less than 4 (the next binary place value), a 0 is placed in the sixth binary digit space: 111010XX. Since 1 is less than 2 (the next binary place value), a 0 is placed in the seventh binary digit space: 1110100X. 1 – 1 = 0 A “1” is placed in the eighth binary digit space: 11101001. The decimal number 233 is equivalent to the binary number 11101001, as shown in Figure 3-36. Three additional decimal number to binary number conversions are shown in Figure 3-36.