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The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values. These values are listed in the following table for angles from 0° to 45°. [ 1 ] In the table below, the label "Undefined" represents a ratio 1 : 0. {\displaystyle 1:0.}
The most popular are trigonometric, usually sine and tangent, common logarithm (log 10) (for taking the log of a value on a multiplier scale), natural logarithm (ln) and exponential (e x) scales. Others feature scales for calculating hyperbolic functions .
In slide rule terminology, "folded" means a scale that starts and finishes at values offset from a power of 10. Often folded scales start at π but may be extended lengthways to, say, 3.0 and 35.0. Folded scales with the code subscripted with "M" start and finish at log 10 e to simplify conversion between base-10 and natural logarithms.
In about 1970 HP co-founder Bill Hewlett challenged France Rode to create a "shirt-pocket sized HP-9100".At the time, slide rules were the only practical portable devices for performing trigonometric and exponential functions, as existing pocket calculators could only perform addition, subtraction, multiplication, and division.
Trigonometric functions were among the earliest uses for mathematical tables. [48] Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. [49] Slide rules had special scales for trigonometric functions. [50]
These two starting trigonometric values are usually computed using existing library functions (but could also be found e.g. by employing Newton's method in the complex plane to solve for the primitive root of z N − 1). This method would produce an exact table in exact arithmetic, but has errors in finite-precision floating-point arithmetic
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
John Napier (1550–1617), the inventor of logarithms Title page of Napier's 1614 table of logarithms of trigonometric functions Mirifici Logarithmorum Canonis Descriptio The 19 degree pages from Napier's 1614 table. The left hand page covers angle increments of 0 to 30 minutes, the right hand page 30 to 60 minutes
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