Property	Value
Hex Value	`$C3`
Categories	Catalog > S Keypad
Localizations	FR: `Arcsin(`

`sin⁻¹(`

Overview

Returns the arcsine of a real number, expression, or list.

Availability: Token available everywhere.

Syntax

sin⁻¹(value)

Arguments

Name	Type	Optional
⁻¹	real\|expression\|real[]
value	real\|expression\|real[]

Location

2nd, sin⁻¹

Description

sinֿ¹( returns the arcsine of its argument. It is the inverse of sin(, which means that sinֿ¹(z) produces an angle θ such that sin(θ)=z.

Like sin(, the result of sinֿ¹( depends on whether the calculator is in Radian or Degree mode. However, unlike sine, the result is in degrees or radians, not the argument. A full rotation around a circle is 2π radians, which is equal to 360°. The conversion of θ=sinֿ¹(n) from radians to degrees is θ180/π and from degrees to radians is θπ/180. The sinֿ¹( command also works on lists.

The sinֿ¹( function can be defined for all real and complex numbers; however, the function assumes real values only in the closed interval [-1,1]. Because the trigonometric functions and their inverses in the Z80 calculators are restricted only to real values, the calculator will throw ERR:DOMAIN if the argument is outside of this interval, no matter what the mode setting may be.

In radians:

:sinֿ¹(1)
    1.570796327

In degrees:

:sinֿ¹(1)
    90

Advanced Uses

Since the function sine itself doesn't have the restrictions that arcsine does, and since arcsine is the inverse of sine, you can use sinֿ¹(sin( to keep a variable within a certain range (most useful on the graph screen). Here is an example for a game like pong. The ball travels between -6 and 6.

You could use a flag like this:

:If 6=abs(X        \\ X is the position
:-D→D        \\ D is the direction
:X+D→X        \\ new position
:Pt-On(-54,X,"=")

An easier way to do this, without needing a flag or even an If statement, is using sinֿ¹(sin(

:X+1→X        \\ Note: the calculator is in degree mode
:Pt-On(-54,sinֿ¹(sin(15X))/15,"=")    \\ 15 is used because sinֿ¹ ranges from [-90,90]
                                        and X from [-6,6],  so 90/6=15

Error Conditions

ERR:DATA TYPE is thrown if you input a complex value or a matrix.
ERR:DOMAIN is thrown if you supplied an argument outside the interval [-1,1]

Source: parts of this page were written by the following TI|BD contributors: burr, DarkerLine, GoVegan, simplethinker, thornahawk.

History

Calculator	OS Version	Description
TI-82	1.0	`sin⁻¹` added
TI-83	0.01013	Renamed `sin⁻¹` to `sin⁻¹(`

Last update: September 30, 2023 15:38:51
Created: February 23, 2023 23:15:01

Authors: Adrien Bertrand

sin⁻¹(