Returns the inverse sine (also called arcsine)

sinֿ¹(*number*)

Press:

- [2nd]
- [sinֿ¹]

TI-83/84/+/SE

1 byte

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 θ*180/π and from degrees to radians is θ*π/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]

# Related Commands

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