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HP 35s Calculator - Understanding Trigonometric Functions in Reverse Polish Notation (RPN)

Introduction
This document pertains to trigonometric functions for the HP 35s calculator.
Entering pi
Press to place the first 12 digits of π into the X–register. The number of digits displayed depends on the display format. Because is a function that returns an approximation of π to the stack, it is not necessary to press .
  note:
A calculator cannot exactly represent π, since π is a transcendental number.
Setting the angular mode
The angular mode specifies which unit of measure to assume for angles used in trigonometric functions. The mode does not convert numbers already present.
360 degrees = 2π radians = 400 gradients
To set an angular mode, press . The 'Mode' menu displays options which may be selected by pressing either the number to the left of the item, or by highlighting the item with the directional arrow keys and pressing with the item underlined.
Option
Description
Annunciator
DEG
Sets degree mode. Uses decimal degrees rather than hexagesimal degrees (degrees, minutes, seconds).
None
RAD
Sets radian mode.
RAD
GRAD
Sets gradient mode.
GRAD
Trigonometric functions in RPN
With x in the display:
Press:
To Calculate:
Sine of x.
Cosine of x.
Tangent of x.
Arc sine of x.
Arc cosine of x.
Arc tangent of x.
  note:
Calculations with the irrational number π cannot be expressed exactly by the 15-digit internal precision of the calculator. This is particularly noticeable in trigonometry. For example, the calculated sin π (radians) is not zero but –2.0676 × 10–13, a very small number close to zero. In Algebraic mode, press the key for the trigonometric function first, followed by x. Press to return results.
Example of trigonometric functions in RPN mode
Show that cosine (5/7)π radians and cosine 128.57° are equal (to four significant digits).
Key
Display
Description
(2RAD)
Sets Radians mode; RAD annunciator on.
0.7143
5/7 in decimal format.
-0.6235
Cos (5/7)π.
(1DEG)
-0.6235
Switches to Degrees mode (no annunciator).
-0.6235
Calculates cos 128.57°, which is the same as cos (5/7)π.
Programming note
Equations using inverse trigonometric functions to determine an angle θ, often look something like this:
θ = arctan (y/x).
If x = 0, then y/x is undefined, resulting in the error DIVIDE BY 0.

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