Machinery's Handbook, 31st Edition
TRIGONOMETRIC FUNCTIONS 111 Using a Calculator to Find Trigonometric Function Values.— Scientific calculators are quicker and more accurate than tables for finding trigonometric ratios or angles than relying on trigonometric tables. Inputting an angle, in either degree ( DEG ) or radian ( RAD ) mea- sure, and pressing SIN , COS , or TAN key will produce the ratio value to a many decimal places, which can be rounded to the desired accuracy. Though reciprocal function keys are not usually included on the calculator, using the main three functions and the 1/x key will produce these ratios as well, since csc x = 1/sin x , sec x = 1/cos x , and cot x = 1/tan x . If the triangle’s side dimensions are known and the angle measure is sought, then ratio is entered using the keys labeled sin –1 , cos –1 , and tan –1 . If these keys are not present, then INV is used with SIN , COS , or TAN . Again, the correct units, whether degree or radian measure, can be chosen. An advantage of using a calculator instead of a trigonometry table to find function values is that both positive and negative degree measures can be entered into a calculator; also, angles greater than 90 degrees do not need to be converted to their acute angle equivalent. Interpolation for angles whose measures fall between the values available in the tables is also not necessary. Example: Use a calculator to find all six of the trigonometry function values, to four deci - mal places, of 172°. Solution: The degree measure can be entered as is, without having to first inspect its quad - rant and reference (acute) angle. Enter 172 in DEG mode, then, for the three main values, choose SIN, COS, TAN, each time rounding answer to four decimal places. sin(172°) = 0.1392 csc(172°) = 1/sin(172°) = 7.1853 cos(172°) = –0.9903 sec(172°) = 1/cos(172°) = –1.0098 tan(172°) = –0.1405 cot(172°) = 1/tan(172°) = –7.1153 If a scientific calculator or computer is not available, tables are the easiest way to find trig values. However, trigonometric function values can be calculated very accurately without a scientific calculator by using the infinite series formulas (see CALCULUS ): A sin A A 3 3! --- A 5 5! --- A 7 7! --- ± … = – + – A sin –1 --- 1 2 A + where angle A is expressed in radians (multiplying degrees by p /180 = 0.0174533 gives radian measure). Generally, calculating just three or four terms of the expression is sufficient for accuracy. In these formulas, a number followed by the symbol ! is called a factorial (see Fac- torial Notation on page 13). Except for 0!, which equals 1, n ! = n ( n – 1)( n – 2) . . . down to 1. For example, 4! = 4 × 3 × 2 × 1 = 24. As an example, sin 42° = sin (42 × 0.0174533) = sin (0.733) = 0.733 – (0.733) 3 /3! + (0.733) 5 /5! – . . . ≈ 0.66912, which is close to the calculator answer of higher accuracy, 0.669130606. Versed Sine and Versed Cosine.— These functions are sometimes used in formulas for segments of a circle and may be obtained using the relationships: versed θ sin 1 θ cos versed θ cos – 1 θ – sin . = = ; = --- A 5 5 7 --- ±… = – + – Sevolute Functions.— Sevolute functions are used in calculating the form diameter of in- volute splines. They are computed by subtracting the involute function of an angle from the secant of the angle (1/cos θ = sec θ ). For example, sevolute of 20 ° = secant of 20 ° - involute function of 20 ° = 1.064178 - 0.014904 = 1.049274. Involute Functions.— Involute functions are used in certain formulas relating to the design and measurement of gear teeth as well as measurement of threads over wires. (See, for exam- ple, pages 2130 through 2133, 2286, and 2350). The value of an involute function is calculated from the following formulas: Involute of θ = (tan θ ) – θ for θ in radians, and Involute of θ = (tan θ ) – π × θ /180 for θ in degrees. Example: For an angle of 14 degrees and 10 minutes (14° 10´), the involute function is found as follows: 10 minutes = 10 ⁄ 60 = 0.166666°, 14 + 0.166666 = 14.166666°, so that the involute of 14.166666° = (tan 14.166666) – π × 14.166666 ⁄ 180 = 0.252420 – 0.247255 = 0.005165. The same result would be obtained by using the conversion tables beginning on page 112 to convert 14° 10´ to radians and then applying the first involute formula for radian measure given above. 1 2 -- A 3 3 × -- 3 4 -- A 5 5 --- … + × × + A cos 1 A 2 2! A tan –1 A A 3 3 --- A 4 4! 6! --- ± … = – + – --- A 6 --- A 7
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