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# Inverses of Circular Functions Help (page 2)

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By McGraw-Hill Professional
Updated on Oct 3, 2011

## Arc What?

We can now define the inverses of the circular functions. There are two ways of denoting an inverse when talking about the sine, cosine, tangent, cosecant, secant, and cotangent. We can use the standard abbreviation and add a superscript –1 after it, or we can write “arc” in front of it. Here are the animals, one by one:

• The inverse of the sine function is the arcsine function. If we are operating on some variable x , the arcsine of x is denoted sin –1 ( x ) or arcsin ( x )
•
• The inverse of the cosine function is the arccosine function. If we are operating on some variable x , the arccosine of x is denoted cos –1 ( x ) or arccos ( x )
• The inverse of the tangent function is the arctangent function. If we are operating on some variable x , the arctangent of x is denoted tan –1 ( x ) or arctan ( x )
• The inverse of the cosecant function is the arccosecant function. If we are operating on some variable x , the arccosecant of x is denoted csc –1 ( x ) or arccsc ( x )
• The inverse of the secant function is the arcsecant function. If we are operating on some variable x , the arcsecant of x is denoted sec –1 ( x ) or arcsec ( x )
• The inverse of the cotangent function is the arccotangent function. If we are operating on some variable x , the arccotangent of x is denoted cot –1 ( x ) or arccot ( x )

The sine, cosine, tangent, cosecant, secant, and cotangent require special restrictions in order for the inverses to be definable as legitimate functions. These limits are shown in the graphs of the inverse functions that follow.

## Use (and Misuse) Of The –1 Superscript

When using –1 as a superscript in trigonometry, we have to be careful. Ambiguity, or even nonsense, can be the result of improper usage. The expression sin –1 x is not the same thing as (sin x ) –1 . The former expression refers to the inverse sine of x , or the arcsine of x (arcsin x ); but the latter expression means the reciprocal of the sine of x , that is, 1/(sin x ). These are not the same. If you have any question about this, plug in a few numbers and test them.

This brings to light an inconsistency in mathematical usage. It is customary to write (sin x ) 2 as sin 2 x . But don’t try that with the exponent –1, for the reason just demonstrated. You might wonder why the numbers 2 and –1 should be treated so much differently when they are used as superscripts in trigonometry. There is no good answer, except that it is “mathematical convention.”

What about other numbers? Does sin –3 x , for example, mean the reciprocal of the cube of the sine of x , or the cube of the arcsine of x ? Or does it mean the arcsine of the cube of x ? If you are worried that the use of a certain notation or expression might produce confusion, don’t use it. Use something else, even if it looks less elegant. Saying what you mean is more important than conservation of symbols. It is better to look clumsy and be clear and correct, than to look slick and be ambiguous or mistaken.

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