Sections View Full Chapter Figures Tables Videos Annotate Full Chapter Figures Tables Videos Supplementary Content + Mathematics Review Download Section PDF Listen ++ 1. Order of Arithmetic OperationsCertain arithmetic operations take precedence over others. In completing problems with a series of operations the following guidelines apply:a. Addition or subtraction may occur in any order.Example: 4 + 8 − 7 + 3 = 8 or 8 + 3 + 4 − 7 = 8b. Multiplication or division must be completed before addition or subtraction.Example: 48 ÷ 6 + 2 = 10Example: 4 + (2/3)(1/2) = 4 1/3c. Any quantity above a division line, under a division line or a radical sign , or within parentheses or brackets must be treated as one number.2. Fractions, Decimals, and Percentsa. To add (or subtract) fractions, the denominator in each term must be the same. (Choose the lowest common denominator for each term. Multiply each term by the common denominator and then add [or subtract].)(lowest common denominator = 12)Solution:(lowest common denominator = xc)Solution:b. To multiply fractions, multiply the numerators by each other and the denominators by each other.c. To divide fractions, invert the divisor and multiply.d. To convert a fraction to a percentage divide the numerator by the denominator and multiply by 100.Note: To convert a percentage to a decimal move the decimal point two places to the left.e. When dividing by a decimal divide by the integer and add sufficient zeros to move the decimal point the appropriate number of digits to the right.(appropriate number of digits to right = 2)When multiplying by a decimal multiply the integer and add enough zeros to move the decimal point the appropriate number of digits to the left.(appropriate number of digits to left = 3)f. Decimals may be expressed as positive or negative powers of 10:3. Proportions, Formulas, and EquationsThe location of values in proportions, equations, or formulas may be shifted provided that whatever addition, subtraction, multiplication, or division is performed on one side of the equation is also performed on the other side.4. Right Triangles and Trigonometric Equationsa. In a right triangle one angle always equals 90°. The other two angles will always be acute angles and the sum of these two angles will be 90° since the sum of the angles in any triangle is 180°.b. In a right triangle the sides are related to each other so that the square of the longest side or hypotenuse (c) is equal to the sum of the squares of the two sides: c2 = a2 + b2. This is the Pythagorean theorem.c. In triangle ABC, side a is called the side opposite angle A, side b is opposite angle B, and the hypotenuse, c, is opposite the right angle. Side b is named the side adjacent to angle A and side a is the side adjacent to angle B.d. Trigonometric functions are ratios between the sides of a right triangle and are determined by the value of one of the acute angles. There are six trigonometric functions—the sine, cosine, tangent, cotangent, secant, and cosecant—but it will be necessary to consider only the first four here.In Δ ABC the ratio between the side opposite one of the acute angles and the hypotenuse is called the sine of the angle. For angle A it would be written as The cosine expresses the ratio between the side adjacent and the hypotenuse. For angleThe tangent and cotangent represent ratios between the two sides of theA glance at these values shows that sin A = cos B, and that tan A = cot B.As can be seen from studying these ratios, two functions may have the same ratio. For instance, the sin A = cos B and the tan A = cot B.In general terms these trigonometric functions are expressed as follows:*θ (Greek letter, theta) is the symbol for angle.e. Value of trigonometric functions may be obtained from tables of trigonometric functions (see Appendix E) or from handheld calculators with trigonometric function capability.Example: sin 60° = .8660cos 30° = .8660tan 22° = .4040cot 68° = .4040Tables of trigonometric functions usually go up to 90°. Angles greater than 90° may be handled as follows:(1) Functions of angles greater than 90° but less than 180° are the same as functions of an angle equal to 180° minus the angle in question. All functions of angles in this range are negative except the sine.Example: sin 120° = sin 60° tan 150° = –tan 30°(2) Functions of angles greater than 180° but less than 270° are the same as functions of an angle equal to 270° minus the angle in question. Functions of angles in this range are negative except for the tan and cot.Example: cot 220° = –cos 50° tan 195° = tan 75°(3) Functions of angles greater than 270° but less than 360° are the same as functions of an angle equal to 360° minus the angle in question. All functions of angles in this range are negative except the cosine.Example: cot 300° = –cot 60° sin 330° = –sin 30°f. Through the use of trigonometric functions, it is possible to determine the values of all components of a triangle when the values of one side and one angle or the values of two sides are known.Example 1: In triangle ABC, angle A = 25° and the length of the hypotenuse is 15 m. Find the length of the other two sides.Solution:Example 2: In triangle ABC the lengths of the sides are 3 cm and 5 cm. What is the length of the hypotenuse and the size of both acute angles?Solution:Note: c may also be found using the Pythagorean theorem: C2 = a2 + b25. Geometry of Circlesa. The circumference of a circle is calculated using the formula C = 2πr, where C is the circumference, r is the radius, and π (pi) is a constant value of 3.1416. Pi is the ratio that exists between the diameter of a circle and its circumference.b. In making one complete turn about a circle the radius goes through one revolution, 360° or 2π radians. A radian is the angle subtended by an arc of a circle equal in length to the radius. One radian equals or 57.3°. Some equivalents for these angular units of measure are as follows:(1) To convert degrees to revolutions divide by 360.Example: 1260° = 3.50 rev(2) To convert radians to revolutions divide by 6.28.Example: 15.75 radians = 2.51 rev(3) To convert degrees to radians divide by 57.3.Example: 360° = 6.28 radians(4) To convert revolutions to radians multiply by 6.28.Example: 2.3 rev = 14.44 radians(5) To convert revolutions to degrees multiply by 360.Example: 2.3 rev = 828°(6) To convert radians to degrees multiply by 57.3.Example: 7.6 radians = 435.5°