# MyReckonings.com

## Dead Reckoning: Calculating Without Instruments

Chapter 5: Trigonometric Functions and Their Inverses

Background and Topics

This chapter provides convenient 3- to 4-digit approximations to the sine, cosine, tangent, arcsine, arccosine and arctangent functions in units of degrees or radians. Graphs provide error curves for each method as a means of evaluating them.

Notes and Errata (If you have any more to contribute, please email me! This is a compendium of feedback)  A printer-friendly summary table for all chapters is found here.

 Color Code Type Meaning Note A clarification or elaboration of the text. Typo A simple mistake that does not affect the method presented. Error An error that may affect a method or the reader interpretation of it.
 Page Code Explanation General It is my preference in this book when expressing a decimal value that continues on past the digits that are displayed, that the last digit shown is rounded off, followed by ellipses (“…”). For example, a value of log 2 = 0.3010299956…when shown to 5 decimal places will be given as 0.30103… rather than 0.30103 or 0.30102… This helps a great deal when comparing an approximation to an exact value to a certain number of places. 147 The value 174 in Equation 22 is different from the rounded value of 174.51 from the previous equation because 174 provides a better approximation over the whole range. The adjustment of this value for overall accuracy is also done in Equation 23 and Equation 27. 148 The initial slope of the sine function should be given as 0.017453. 151-152 I say that the sine approximation is valid in the range where the cosine approximation is invalid, and vice-versa. The next two sentences show what I really meant there--that the range of angles in which the sine approximation is valid (0-54 degrees) and the range of angles in which the cosine approximation is valid (0-36 degrees) are convenient because for a>54, we can replace sin(a) with cos(90-a) and watch the sign on the answer, and for b>36, we can replace cos(b) with sin(90-b). 152 In the equation that merges Equations 23 and 24, the coefficient would actually be 171.43, but is given as 172 to match the formula given by Ozanam. 153 The value 0.53168 for tan 28º would be 0.53166 if the sine and cosine values are taken only to the 4 decimal places given in the earlier calculations. 154 The reference to Equation 23 should be to Equation 22. 156 In the top equation, “28x30/40” should read “28x30/20”. The result is correct. To get the result 0.53145 for tan 28º, tan 30º must be known to 5 decimal places. The intent here is to find the ultimate accuracy possible from the formula. New methods of approximating the tangent function and the hyperbolic tangent function can be found in the paper Fast Approximation of the Tangent, Hyperbolic Tangent, Exponential and Logarithmic Functions described and linked in the Additional Materials section below. 157 In Equation 33, the sign on .5 should be reversed. The caption for Figure 6 should indicate that the sign of the y-value is reversed for the plot from Equation 33 to show the intersection of the absolute values of the errors of the two plots. 158 The phrase “dropping a b^4 term” should read “dropping a b^3 term”. 160 The caption for Figure 8 should indicate that the sign of the y-value is reversed for the plot optimized for x<=.707 to show the intersection of the absolute values of the errors of the two plots. 161 One of the g subscripts is a letter “o” instead of a number “0”. 165 The arctanh formula should use ln.

Additional Materials Related to Topics in This Chapter

Fast Approximation of the Tangent, Hyperbolic Tangent, Exponential and Logarithmic Functions: A few methods of calculating the tangent function are given in the book, but the one most conducive to mental calculation provides only three-digit accuracy, even with the correction added from the table provided.  This paper derives new methods of mine that provide fast calculations of the tangent and hyperbolic tangent functions to an accuracy of essentially four digits.  (It is also my first paper formatted using the LaTeX mathematical typesetting language.)

Cited Reference Materials

In the near future, this section will contain relevant sections of some of the references cited at the end of this chapter.