## Arc and also arc functions

**An arc role undoes a trig or hyperbolic trig function.**

You are watching: Sin^-1 (1/2)

You are watching: Sin^-1 (1/2)

strictly speaking, the symbol sin-1( ) or Arcsin( ) is provided for the Arcsine function, the function that undoes the sine. This role returns only one answer for each input and it synchronizes to the blue arcsine graph in ~ the left.

Arcsine might be believed of as "the angle who sine is" make arcsine(1/2) average "the angle whose sine is 1/2" or /6.

The symbol sin-1( ) is used frequently when one wishes an ext than one or also all the values feasible even though this values are not spanned by the Arcsine function. See listed below for a much better understanding that this.

Think of the Arcsine as the principal arcsine.

Restrict the domain to take it this train station function.A function can only be an inverse if that is 1-to-1 and also undoes specifically the wanted function. Check out inverse role notes because that a evaluation of train station functions.

In the graph at the left, notification that the sine function, pink and also dashed, is not 1-to-1 because it is periodic and repeats every 2.

The only method for f(x) = sin(x) to undo g(x)= sin-1(x) is if that is 1-to-1 which needs the domain to be restricted.

once this is excellent f(x) = sin(x) undoes g(x)= sin-1(x) and also g(x)= sin-1(x) undoes f(x) = sin(x).

If we only use the pink part, from - to +, then once the minimal sine is reflected end the heat y=x to take it the train station graphically, the inverse, g(x) = sin-1(x) is found and also will indeed give us a single value for every x value from -1 come 1, inclusive.

This restriction provides the domain of the Arcsine - x + and the selection to -1 y 1 together is needed.

**Exactly how Does the Work?**

The arcsine the a positive number is a first quadrant angle, sin-1(+) is in quadrant I.** The arcsine of zero is zero, sin-1(0) is 0. The arcsine that a an unfavorable number is a negative first quadrant angle, sin-1(-) is in quadrant -I,a clockwise-angle of much less than or equal to -/2.**

** The arccosine of a optimistic number is a very first quadrant angle, cos-1(+) is in quadrant I. The arccosine that zero is /2, cos-1(0) is /2. The arccosine that a an unfavorable number is a second quadrant angle, cos-1(-) is in quadrant II.**

** The arctangent of a positive number is a very first quadrant angle, tan-1(+) is in quadrant I. The arctangent that zero is zero, tan-1(0) is 0. The arctangent the a an unfavorable number is a negative first quadrant angle, sin-1(-) is in quadrant -I,a clockwise-angle of much less than - /2.**

**When you simplify an expression, be certain to usage the Arcsine.**

**Simplify.**

When you fix an equation, be responsibility of the domain of the x in the equation and also how plenty of solutions you need to be looking for.

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Answers. | |

1. Arcsin(1/2) | arcsin(1/2)= /6, 30°, clearly in the range of the Arcsine function |

2. Arcsin(-1/2) | arcsin(-1/2)= -/6, -30°,clearly in the range of the Arcsine function. perform not usage the 4th quadrant edge 11/6, 330°, also though the sine the 330° is -1/2, because 11/6, is no in the range of the Arcsine function. |

3. Arcsin(sin(x)) | x, one duty undoes the other |

4. Sin(arcsin(x)) | x, one role undoes the other |

5. Arcsin(sin(30°)) | 30°, the angle who sine is the sine that 30° is 30° |

6. Arcsin(sin(210°)) | arcsin(sin(210°))=arcsin(-1/2)=-/6, -30° due to the fact that the answer must be in the selection of the arcsine function |

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though the Arcsine role is offered to deal with the equation, services to the equation might not it is in in the range of the arcsine function.

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