Sách lượng giác/Hàm số lượng giác/Hàm lượng giác cơ bản nghịch

Hàm lượng giác cơ bản nghịch sửa

Name Usual notation Definition Domain of   for real result Range of usual principal value
(radians)
Range of usual principal value
(degrees)
arcsine   x = sin(y)      
arccosine   x = cos(y)      
arctangent   x = tan(y) all real numbers    
arccotangent   x = cot(y) all real numbers    
arcsecant   x = sec(y)      
arccosecant   x = csc(y)      

Relationships among the inverse trigonometric functions sửa

 
The usual principal values of the arcsin(x) (red) and arccos(x) (blue) functions graphed on the cartesian plane.
 
The usual principal values of the arctan(x) and arccot(x) functions graphed on the cartesian plane.
 
Principal values of the arcsec(x) and arccsc(x) functions graphed on the cartesian plane.

Complementary angles:

 

Negative arguments:

 

Reciprocal arguments:

 

Useful identities if one only has a fragment of a sine table:

 

Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real).

A useful form that follows directly from the table above is

 .

It is obtained by recognizing that  .

From the half-angle formula,  , we get:

 

Arctangent addition formula sửa

 

This is derived from the tangent addition formula

 

by letting

 

Derivatives of inverse trigonometric functions sửa

The derivatives for complex values of z are as follows:

 

Only for real values of x:

 

For a sample derivation: if  , we get:

 

Expression as definite integrals sửa

Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:

 

When x equals 1, the integrals with limited domains are improper integrals, but still well-defined.

Infinite series sửa

Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivative,  , as a binomial series, and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative   in a geometric series, and applying the integral definition above (see Leibniz series).

 
 

Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example,  ,  , and so on. Another series is given by:

 

Leonhard Euler found a series for the arctangent that converges more quickly than its Taylor series:

 

(The term in the sum for n = 0 is the empty product, so is 1.)

Alternatively, this can be expressed as

 

Another series for the arctangent function is given by

 

where   is the imaginary unit.

Continued fractions for arctangent sửa

Two alternatives to the power series for arctangent are these generalized continued fractions:

 


Continued fractions for arctangent sửa

Two alternatives to the power series for arctangent are these generalized continued fractions:

 


Indefinite integrals of inverse trigonometric functions sửa

For real and complex values of z:

 

For real x ≥ 1:

 

For all real x not between -1 and 1:

 

The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecant functions. The signum function is also necessary due to the absolute values in the derivatives of the two functions, which create two different solutions for positive and negative values of x. These can be further simplified using the logarithmic definitions of the inverse hyperbolic functions:

 

The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical to the signum logarithmic function shown above.

All of these antiderivatives can be derived using integration by parts and the simple derivative forms shown above.

Example sửa

Using   (i.e. integration by parts), set

 

Then

 

which by the simple substitution   yields the final result:

 

Logarithmic forms sửa

These functions may also be expressed using complex logarithms. This extends their domains to the complex plane in a natural fashion. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts.

 

Generalization sửa

Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using Euler's formula to form a right triangle in the complex plane. Algebraically, this gives us:

 

or

 

where   is the adjacent side,   is the opposite side, and   is the hypotenuse. From here, we can solve for  .

 

or

 

Simply taking the imaginary part works for any real-valued   and  , but if   or   is complex-valued, we have to use the final equation so that the real part of the result isn't excluded. Since the length of the hypotenuse doesn't change the angle, ignoring the real part of   also removes   from the equation. In the final equation, we see that the angle of the triangle in the complex plane can be found by inputting the lengths of each side. By setting one of the three sides equal to 1 and one of the remaining sides equal to our input  , we obtain a formula for one of the inverse trig functions, for a total of six equations. Because the inverse trig functions require only one input, we must put the final side of the triangle in terms of the other two using the Pythagorean Theorem relation

 

The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for   that result from plugging the values into the equations above and simplifying.

 

In order to match the principal branch of the natural log and square root functions to the usual principal branch of the inverse trig functions, the particular form of the simplified formulation matters. The formulations given in the two rightmost columns assume   and  . To match the principal branch   and   to the usual principal branch of the inverse trig functions, subtract   from the result   when  .

In this sense, all of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since these definition work for any complex-valued  , the definitions allow for hyperbolic angles as outputs and can be used to further define the inverse hyperbolic functions. Elementary proofs of the relations may also proceed via expansion to exponential forms of the trigonometric functions.

Example proof sửa

 

Using the exponential definition of sine, and letting  

 

(the positive branch is chosen)

 
Color wheel graphs of inverse trigonometric functions in the complex plane
     
     
     
     

Expression as definite integrals sửa

Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:

 

When x equals 1, the integrals with limited domains are improper integrals, but still well-defined.

Infinite series sửa

Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivative,  , as a binomial series, and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative   in a geometric series, and applying the integral definition above (see Leibniz series).

 
 

Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example,  ,  , and so on. Another series is given by:

 

Leonhard Euler found a series for the arctangent that converges more quickly than its Taylor series:

 

(The term in the sum for n = 0 is the empty product, so is 1.)

Alternatively, this can be expressed as

 

Another series for the arctangent function is given by

 

where   is the imaginary unit.

Continued fractions for arctangent sửa

Two alternatives to the power series for arctangent are these generalized continued fractions:

 


Continued fractions for arctangent sửa

Two alternatives to the power series for arctangent are these generalized continued fractions:

 


Indefinite integrals of inverse trigonometric functions sửa

For real and complex values of z:

 

For real x ≥ 1:

 

For all real x not between -1 and 1:

 

The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecant functions. The signum function is also necessary due to the absolute values in the derivatives of the two functions, which create two different solutions for positive and negative values of x. These can be further simplified using the logarithmic definitions of the inverse hyperbolic functions:

 

The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical to the signum logarithmic function shown above.

All of these antiderivatives can be derived using integration by parts and the simple derivative forms shown above.

Example sửa

Using   (i.e. integration by parts), set

 

Then

 

which by the simple substitution   yields the final result:

 

Logarithmic forms sửa

These functions may also be expressed using complex logarithms. This extends their domains to the complex plane in a natural fashion. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts.

 

Generalization sửa

Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using Euler's formula to form a right triangle in the complex plane. Algebraically, this gives us:

 

or

 

where   is the adjacent side,   is the opposite side, and   is the hypotenuse. From here, we can solve for  .

 

or

 

Simply taking the imaginary part works for any real-valued   and  , but if   or   is complex-valued, we have to use the final equation so that the real part of the result isn't excluded. Since the length of the hypotenuse doesn't change the angle, ignoring the real part of   also removes   from the equation. In the final equation, we see that the angle of the triangle in the complex plane can be found by inputting the lengths of each side. By setting one of the three sides equal to 1 and one of the remaining sides equal to our input  , we obtain a formula for one of the inverse trig functions, for a total of six equations. Because the inverse trig functions require only one input, we must put the final side of the triangle in terms of the other two using the Pythagorean Theorem relation

 

The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for   that result from plugging the values into the equations above and simplifying.

 

In order to match the principal branch of the natural log and square root functions to the usual principal branch of the inverse trig functions, the particular form of the simplified formulation matters. The formulations given in the two rightmost columns assume   and  . To match the principal branch   and   to the usual principal branch of the inverse trig functions, subtract   from the result   when  .

In this sense, all of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since these definition work for any complex-valued  , the definitions allow for hyperbolic angles as outputs and can be used to further define the inverse hyperbolic functions. Elementary proofs of the relations may also proceed via expansion to exponential forms of the trigonometric functions.

Example proof sửa

 

Using the exponential definition of sine, and letting  

 

(the positive branch is chosen)

 
Color wheel graphs of inverse trigonometric functions in the complex plane
     
     
     
     

Expression as definite integrals sửa

Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:

 

When x equals 1, the integrals with limited domains are improper integrals, but still well-defined.

Infinite series sửa

Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivative,  , as a binomial series, and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative   in a geometric series, and applying the integral definition above (see Leibniz series).

 
 

Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example,  ,  , and so on. Another series is given by:

 

Leonhard Euler found a series for the arctangent that converges more quickly than its Taylor series:

 

(The term in the sum for n = 0 is the empty product, so is 1.)

Alternatively, this can be expressed as

 

Another series for the arctangent function is given by

 

where   is the imaginary unit.

Continued fractions for arctangent sửa

Two alternatives to the power series for arctangent are these generalized continued fractions:

 


Continued fractions for arctangent sửa

Two alternatives to the power series for arctangent are these generalized continued fractions:

 


Indefinite integrals of inverse trigonometric functions sửa

For real and complex values of z:

 

For real x ≥ 1:

 

For all real x not between -1 and 1:

 

The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecant functions. The signum function is also necessary due to the absolute values in the derivatives of the two functions, which create two different solutions for positive and negative values of x. These can be further simplified using the logarithmic definitions of the inverse hyperbolic functions:

 

The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical to the signum logarithmic function shown above.

All of these antiderivatives can be derived using integration by parts and the simple derivative forms shown above.

Example sửa

Using   (i.e. integration by parts), set

 

Then

 

which by the simple substitution   yields the final result:

 

Logarithmic forms sửa

These functions may also be expressed using complex logarithms. This extends their domains to the complex plane in a natural fashion. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts.

 

Generalization sửa

Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using Euler's formula to form a right triangle in the complex plane. Algebraically, this gives us:

 

or

 

where   is the adjacent side,   is the opposite side, and   is the hypotenuse. From here, we can solve for  .

 

or

 

Simply taking the imaginary part works for any real-valued   and  , but if   or   is complex-valued, we have to use the final equation so that the real part of the result isn't excluded. Since the length of the hypotenuse doesn't change the angle, ignoring the real part of   also removes   from the equation. In the final equation, we see that the angle of the triangle in the complex plane can be found by inputting the lengths of each side. By setting one of the three sides equal to 1 and one of the remaining sides equal to our input  , we obtain a formula for one of the inverse trig functions, for a total of six equations. Because the inverse trig functions require only one input, we must put the final side of the triangle in terms of the other two using the Pythagorean Theorem relation

 

The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for   that result from plugging the values into the equations above and simplifying.

 

In order to match the principal branch of the natural log and square root functions to the usual principal branch of the inverse trig functions, the particular form of the simplified formulation matters. The formulations given in the two rightmost columns assume   and  . To match the principal branch   and   to the usual principal branch of the inverse trig functions, subtract   from the result   when  .

In this sense, all of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since these definition work for any complex-valued  , the definitions allow for hyperbolic angles as outputs and can be used to further define the inverse hyperbolic functions. Elementary proofs of the relations may also proceed via expansion to exponential forms of the trigonometric functions.

Example proof sửa

 

Using the exponential definition of sine, and letting  

 

(the positive branch is chosen)

 
Color wheel graphs of inverse trigonometric functions in the complex plane
     
     
     
     

Expression as definite integrals sửa

Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:

 

When x equals 1, the integrals with limited domains are improper integrals, but still well-defined.

Infinite series sửa

Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivative,  , as a binomial series, and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative   in a geometric series, and applying the integral definition above (see Leibniz series).

 
 

Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example,  ,  , and so on. Another series is given by:[1]

 

Leonhard Euler found a series for the arctangent that converges more quickly than its Taylor series:

 [2]

(The term in the sum for n = 0 is the empty product, so is 1.)

Alternatively, this can be expressed as

 

Another series for the arctangent function is given by

 

where   is the imaginary unit.[3]

Continued fractions for arctangent sửa

Two alternatives to the power series for arctangent are these generalized continued fractions: