Home | About Mathguru | Advertisements | Teacher Zone | FAQs | Contact Us | Login

 
If you like what you see in Mathguru
Subscribe Today
For 12 Months
US Dollars 12 / Indian Rupees 600
Available in 20 more currencies if you pay with PayPal.
Buy Now
No questions asked full moneyback guarantee within 7 days of purchase, in case of Visa and Mastercard payment
  

Example:Solve Inverse Trigonometric Function

Post to:

Bookmark and Share



Explanation:

 

Inverse trigonometric functions

 

In mathematics, the inverse trigonometric functions or cyclometric functions are the inverse functions of the trigonometric functions, though they do not meet the official definition for inverse functions as their ranges are subsets of the domains of the original functions. Since none of the six trigonometric functions are one-to-one (by failing the horizontal line test), they must be restricted in order to have inverse functions.

For example, just as the square root function  is defined such that y2 = x, the function y = arcsin(x) is defined so that sin(y) = x. There are multiple numbers y such that sin(y) = x; for example, sin (0) = 0, but also sin(π) = 0, sin(2π) = 0, etc. It follows that the arcsine function is multivalued: arcsin (0) = 0, but also arcsin (0) = π, arcsin (0) = 2π, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x in the domain the expression arcsin(x) will evaluate only to a single value, called its principal value. These properties apply to all the inverse trigonometric functions.

The principal inverses are listed in the following table.

Name

Usual notation

Definition

Domain of x for real result

Range of usual principal value
(radians)

Range of usual principal value
(degrees)

arcsine

y = arcsin x

x = sin y

−1 ≤ x ≤ 1

−π/2 ≤ y ≤ π/2

−90° ≤ y ≤ 90°

arccosine

y = arccos x

x = cos y

−1 ≤ x ≤ 1

0 ≤ y ≤ π

0° ≤ y ≤ 180°

arctangent

y = arctan x

x = tan y

all real numbers

−π/2 < y < π/2

−90° < y < 90°

arccotangent

y = arccot x

x = cot y

all real numbers

0 < y < π

0° < y < 180°

arcsecant

y = arcsec x

x = sec y

x ≤ −1 or 1 ≤ x

0 ≤ y < π/2 or π/2 < y ≤ π

0° ≤ y < 90° or 90° < y ≤ 180°

arccosecant

y = arccsc x

x = csc y

x ≤ −1 or 1 ≤ x

−π/2 ≤ y < 0 or 0 < y ≤ π/2

-90° ≤ y < 0° or 0° < y ≤ 90°

(Our solved example in mathguru.com uses this concept)

 

The notations sin−1, cos−1, etc. are often used for arcsin, arccos, etc., but this convention logically conflicts with the common semantics for expressions like sin2(x), which refer to numeric power rather than function composition, and therefore may result in confusion between multiplicative inverse and compositional inverse.

 

http://en.wikipedia.org/wiki/Inverse_trigonometric_functions

 

The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.