Antiderivative Trig

Antiderivative Trig Antiderivative Trig The definition of Antiderivative is, ∫ g (x) dx = f(x )+ c, where d f(x) /dx = g(x). The following are the integrals of the trigonometric functions. Following are the integrals or antiderivatives of the sin, cos, tan, cot, cosec etc. functions.

For general if the sin x is a trigonometric function then cos x is the derivative of that function.

Antiderivatives of some of the sin function are as follows- ∫ sin ax = - (1/a) cos ax + c ∫ sin n ax dx = - sin (n-1) . ax . cos ax / na + (n-1 )/ n . ∫ sin n-2 ax dx cos ax / na + (n-1 )/ n . ∫ sin n-2 ax dx ( for n > 2 ) Antiderivative of the two cos functions are as following- ∫ cos ax dx = (1/a) sin ax +c ∫ cosn ax dx = cos n-1 ax . sin ax / na + ( n-1 / n ) . ∫ cos n-2 ax dx (for n > 0) Integral of the tan x is defined as the formula below ∫ tan ax dx = -( 1 / a ) log ( cos ax ) + c Antiderivative of the \secant function is defined as follows- ∫ sec ax dx = ( 1 / a ) log (sec ax + tan ax ) + c Integral of the cosec x is as follows- ∫ cosec ax dx = -( 1 / a ) log (cosec ax + cot ax ) + c Integral of the cot x is as follows- ∫ cot ax dx = ( 1 / a ) sin ax +c Here c is the integral constant in all the trigonometric functions defined above and a is a constant and n is the positive integral.

Know More About Trigonometry Worksheet

Tutorcircle.com

Page No. : ­ 1/4

In mathematics, the inverse trigonometric functions (occasionally called cyclometric functions) are the inverse functions of the trigonometric functions with suitably restricted domains .

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.

In computer programming languages the functions arcsin, arccos, arctan, are usually called asin, acos, atan. Many programming languages also provide the two-argument atan2 function, which computes the arctangent of y / x given y and x, but with a range of (−π, π].

Since none of the six trigonometric functions are one-to-one, they must be restricted in order to have inverse functions. Therefore the ranges of the inverse functions are subsets of the domains of the original 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.

Read More About Subtracting Mixed Numbers From Whole Numbers

Tutorcircle.com

Page No. : ­ 2/4

Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of 2π.

Sine and cosecant begin their period at 2πk − π/2 (where k is an integer), finish it at 2πk + π/2, and then reverse themselves over 2πk + π/2 to 2πk + 3π/2.

Cosine and secant begin their period at 2πk, finish it at 2πk + π, and then reverse themselves over 2πk + π to 2πk + 2π. Tangent begins its period at 2πk − π/2, finishes it at 2πk + π/2, and then repeats it (forward) over 2πk + π/2 to 2πk + 3π/2.

Cotangent begins its period at 2πk, finishes it at 2πk + π, and then repeats it (forward) over 2πk + π to 2πk + 2π.

Tutorcircle.com

Page No. : ­ 2/3 Page No. : ­ 3/4

Thank You

TutorCircle.com