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3.5. Insert trigonometric functions

 

Basic trigonometric functions

sin Sine asin Inverse sine
cos Cosine acos Inverse cosine
tan Tangent atan Inverse tangent

 

The basic trigonometric functions can be defined in terms of a right triangle. For the angle x at one apex of the right triangle the functions can be defined by:

The sine function, denoted by sin, is de?ned as:

sin(x) = opposite / hypotenuse

The cosine function, denoted by cos, is de?ned as:

cos(x) = adjacent / hypotenuse

The tangent function, denoted by tan, is de?ned as:

tan(x) = opposite / adjacent = sin(x) / cos(x)

Syntax

cos(x)

cos x

:cos x

Examples

cos(1) = 0.540

If the angle unit is set to degrees: cos(45) = 0.707.

If the angle unit is set to grads: cos(45) = 0.760.

 

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.

The following table gives the inverse trigonometric function (principal value branches) along with their domains and ranges.

Function
Name
Abbreviation
Functions
Alternate
Notations
Domain of x Range of
Principal Values

arcsine y = arcsin x y = asin x -1 ≤ x ≤ 1 -π/2 ≤ y ≤ π/2
arccosine y = arccos x y = acos x -1 ≤ x ≤ 1 0 ≤ y ≤ π
arctangent y = arctan x y = atan x -∞ < x < ∞ -π/2 < y < π/2

 

Syntax

arccos(x)

arccos x

:arccos x

acos(x)

acos x

:acos x

x is the cosine of the angle you want and must be from -1 to 1.

Examples

acos(0.5) = 1.0472 radians

If the angle unit is set to degrees: acos(0.5) = 60 degrees.

If the angle unit is set to grads: acos(0.5) = 66.7 grads.

 

Reciprocal trigonometric functions

sec Secant asec Inverse secant
csc Cosecant acsc Inverse cosecant
cot Cotangent acot Inverse cotangent

 

The cosecant csc(x), or cosec(x), is the reciprocal of sin(x), i.e. the ratio of the length of the hypotenuse to the length of the opposite side:

csc(x) = hypotenuse / opposite = 1 / sin(x)

The inverse cosecant satisfies

acsc(x) = asin(1/x)

The secant sec(A) is the reciprocal of cos(A), i.e. the ratio of the length of the hypotenuse to the length of the adjacent side:

sec(x) = hypotenuse / adjacent = 1 / cos(x)

The inverse secant satisfies

asec(x) = acos(1/x)

cotangent

cot(x) = adjacent / opposite = cos(x) / sin(x) = 1 / tan(x)

Syntax

csc(x)

csc x

:csc x

cosec(x)

cosec x

:cosec x

x is the cosine of the angle you want and must be from -1 to 1.

Examples

csc(30) = -1.01211335307018

If the angle unit is set to degrees: csc(30) = 2.

If the angle unit is set to grads: csc(30) = 2.20268926458527.

 

Inverse reciprocal trigonometric functions

The following table gives the inverse trigonometric function (principal value branches) along with their domains and ranges.

Function
Name
Abbreviation
Functions
Alternate
Notations
Domain of x Range of
Principal Values

arcsecant y = arcsec x y = asec x -∞ < x < ∞ 0 ≤ y < π/2 or π/2 < y ≤ π
arccosecant y = arccsc x y = acsc x -∞ < x < ∞ -π/2 ≤ y < 0 or 0 < y ≤ π/2
arccotangent y = arccot x y = acot x -∞ < x < ∞ -π/2 < y < 0 or 0 < y ≤ π/2

 

Syntax

acsc(x)

acsc x

:acsc x

acosec(x)

acosec x

:acosec x

Examples

acsc(1) = 1.5707963267949

If the angle unit is set to degrees: acsc(1) = 90 degrees.

If the angle unit is set to grads: acsc(1) = 100 grads.

 

Hyperbolic functions

sinh Hyperbolic sine asinh Inverse hyperbolic sine
cosh Hyperbolic cosine acosh Inverse hyperbolic cosine
tanh Hyperbolic tangent atanh Inverse hyperbolic tangent
coth Hyperbolic cotangent acoth Inverse hyperbolic cotangent

 

The hyperbolic functions have similar names to the trigonmetric functions, but they are de?ned in terms of the exponential function. Unlike trigonmetric functions, hyperbolic functions are not periodic!

The hyperbolic sine, denoted sinh, is de?ned as:

sinh(x) = (ex - e-x) / 2

The hyperbolic cosine, denoted cosh, is de?ned as:

cosh(x) = (ex + e-x) / 2

The hyperbolic tangent, denoted tanh, is de?ned as:

tanh(x) = (ex - e-x) / (ex + e-x) = sinh(x) / cosh(x)

The hyperbolic cotangent, denoted coth, is de?ned as:

coth(x) = (ex + e-x) / (ex - e-x) = cosh(x) / sinh(x)

The hyperbolic secant, and hyperbolic cosecant functions are not included in c4c but you can simply calculate these yourself

hyperbolic secant (x) = 1 / cosh (x)

hyperbolic cosecant (x) = 1 / sinh (x)

 

Syntax

sinh(x)

cosh x

:tanh x

asinh(x)

acosh x

:atanh x

Examples

sinh(1) = 1.5707963267949
sinh(90) = 6.1020164715892e+038
cosh(1) = 1.54308063481524
cosh(90) = 6.1020164715892e+038

The required number argument x can be any valid numeric expression and changing the mode to DEG or GRAD will have no effects.

 

 


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