Trigonometric Functions


Welcome to trigonometric functions. Here we discuss the mathematical functions of angles, often simply called trig functions. Apart from similarity, the shape of a polygon with three edges and three vertices, a triangle, is completely determined by the angles, because similar triangles keep the same ratios between their sides. As the angles of a triangle add up to 180°, in a triangle which has one angle of 90 degrees, the sum of the two acute angles is 90 degrees. In other words, the third angle is fixed, provided that the two other angles are known.

Moreover, if we know the angles, then the ratios of the sides are determined, irrespective of the overall size of the triangle. And in case the length of any single side is known, then the other two sides can be calculated. It follows that all other angles and lengths are determined, once the value of a single angle and the length of a single side and are known. The aforementioned ratios are given by the trigonometric functions which you can find below. The following depiction of a triangle shows the hypotenuse h, the adjacent side a as well as the opposite side o:

Triangle

Frequent calculations on trigonometricfunctions.com include, for example:

Sine

The sine of an angle is defined as the ratio opposite/hypotenuse, and for the sine the abbreviation sin is the rule.

sin A = o / h

The inverse of the trigonometric function sin is the arcsine, and the reciprocal of the sin is the cosecant, both of which you can find by reading on this page.

Cosine

The cosine of an angle is defined as the ratio adjacent/hypotenuse, and for the cosine the abbreviation cos is common.

cos A = a / h

The inverse of the cosine sin is the arccosine, and the reciprocal of the cos function is called secant. Keep reading this article to learn about both of them.

Tangent

The tangent of an angle is defined as the ratio opposite/adjacent, and for the tangent the abbreviation tan is prevalent, but tg can also be seen.

tan A = o / a

The inverse of the trigonometric function known as tan is the arctangent, and the reciprocal of the tan is the cotangent. They, too, are explained further below.

Cotangent

The cotangent of an angle is defined as adjacent/opposite, and for the cotangent the abbreviation cot is the most frequent. The short forms ctg, cotg and cotan can also be found.

cot A = a / o

The inverse of the cotangent is the arccotangent, and the reciprocal of the cot is the tangent function. Both, the tangent as well as the arccotangent are discussed on this page.

Secant

The secant of an angle is defined as hypotenuse/adjacent, and for the secant the abbreviation sec is common.

sec A = h / a

The inverse function of the secant is the arcsecant, and the reciprocal of the sec is the cosine. You can find about the arcsecant in the section inverse trigonometric functions.

Cosecant

The cosecant of an angle is defined as hypotenuse/opposite, and for the cosecant the abbreviation csc is the rule.

csc A = h / o

The inverse of the cosecant is the arccosecant, and the reciprocal of the csc is the sine function. To learn about the cosecant read our paragraph inverse trigonometric functions further down.

Unit Circle Trigonometric Functions

As the above right-angled triangle definitions for trigonometric functions only work for angles between 0 and 90 degrees, we introduce the unit circle definition for trigonometric functions to account for all negative and positive arguments.

In a Cartesian coordinate system, x2 + y2 = r2 is the origin equation for a circle centered at the origin. If we make a line through the origin O of the circle and the point P, in the intersection of the circle and the line, P(x,y), x = r cos(θ) and y = r sin(θ); θ is the angle formed by OP and the x-axis.

With r = 1 we obtain x = cos(θ) and y = sin(θ):

Unit-circle Definitions Trigonometric Functions
So far, for the ease of the triangle definition of the trig functions, we have been measuring the plane angles in the unit degree, denoted by °. However, the standard unit of angular measure is the radian with the symbol rad.

90° are equal to π/2 for example. To change degrees to the unit radian multiply the angle by π / 180°, and if want to change radians to degrees multiply the argument by 180° / π.

Period of Trigonometric Functions

All the six trigonometric functions are periodic, with a period of 360 degrees / 2π radians except for the tangent and cotangent functions of 180° / π radian. Colloquially, this is refereed to as period of trig functions.

In the next section we discuss the inverse trigonometric functions, which are not periodic in the strict sense. Rather, in order for sin, cos, tan etc having an inverse function, the domains of the original functions are restricted to a subset.

Inverse Trigonometric Functions

The inverse trigonometric functions are the inverse of the functions discussed above with their domains suitably restricted domains. They are often called inverse trig functions, and used to obtain the angle from any of the angle’s trigonometric ratios sin, cos, tan, cot, sec, and csc.

Arcsine

The arcsine is the inverse function of the sine, defined as x = sin(y), usually written as y = arcsin(x). Its domain of x for a real result of arcsin(x) is −1 ≤ x ≤ 1. The range of the usual principal value in degrees is: −90° ≤ y ≤ 90°; in radians −π/2 ≤ y ≤ π/2.

Arccosine

The arccosine is the inverse function of the cosine, defined as x = cos(y), normally notated as y = arccos(x). Its domain of x for a real result of arccos(x) is −1 ≤ x ≤ 1. The range of the usual principal value in degrees is 0° ≤ y ≤ 180°; in radians 0 ≤ y ≤ π.

Arctangent

The arctangent is the inverse function of the tangent, defined as x = tan(y), conventionally spelled as y = arctan(x). The domain of x for a real result of arctan(x) is $\mathbb{R}$, the set of real numbers. Its range of the usual principal value in degrees is −90° < y < 90°; in radians

Arccotangent

The arccotangent is the inverse function of the cotangent, defined as x = cot(y), commonly written as y = arccot(x). Its domain of x for a real result of arccot(x) is $\mathbb{R}$. The range of the usual principal value in degrees is 0° < y < 180°; in radians −π/2 < y < π/2.

Arcsecant

The arcsecant is the inverse function of the secant, defined as x = sec(y), typically notated as y = arcssec(x). The domain of x for a result in $\mathbb{R}$ is x ≤ −1 or 1 ≤ x. Its range of the usual principal value in degrees is 0° ≤ y < 90° or 90° < y ≤ 180°; in radians 0 < y < π.

Arccosecant

The arccosecant is is the inverse function of the cosecant, defined as x = csc(y), generally spelled as y = arccsc(x). Its domain of x for a result in $\mathbb{R}$ is x ≤ −1 or 1 ≤ x. The range of the usual principal value in degrees is −90° ≤ y < 0° or 0° < y ≤ 90°; in radians −π/2 ≤ y < 0 or 0 < y ≤ π/2.

Make sure to understand the difference between the inverse trig functions, just discussed, and the reciprocal of sin cos tan, which is ahead.

Reciprocal Trigonometric Functions

In trigonometry, the inverse functions, and the reciprocal trigonometric functions are not the same: f(x)-1 ≠ 1/f(x). The reciprocal trigonometric functions are as follows:

  • f(x) = sin x, sin-1 x = 1 / csc x
  • f(x) = cos x, cos-1 x = 1 / sec x
  • f(x) = tan x, tan-1 x = 1 / cot x
  • f(x) = cot x, cot-1 x = 1 / tan x
  • f(x) = sec x, sec-1 x = 1 / cos x
  • f(x) = csc x, csc-1 x = 1 / sin x

 

You have made it to the end of our introduction to the trigonometric functions sin cos tan etc. We plan to add an article for each trigonometric function, along with a calculator.

Until then, use our search form located in the sidebar to locate and calculate any trig functions and inverse trig functions. On mobile devices the form is located near the end of this article.

We hope you like our content and share it with your friends using the sharing tools at your disposal. For questions and comments use the form at the bottom, or send us a mail.

Thanks for visiting our site dedicated to trigonometry.

 
Further Information:

Trigonometry on Wikipedia
Trigonometric_functions on Wikipedia

Unit-circle definitions trigonometric functions Image reference: Webysther / Wikimedia.

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