Ask the Chatbot Games & Quizzes History & Society Science & Tech Biographies Animals & Nature Geography & Travel Arts & Culture ProCon Money Videos
trigonometric function
References & Edit History Quick Facts & Related Topics
Images
Diagram of a right triangle labeled with sides a, b, and hypotenuse c, alongside formulas for the six trigonometric functions-sine, cosine, tangent, cosecant, secant, and cotangent-each defined by ratios of the triangle's sides. Diagram of a right triangle with a yellow interior, green outline, green angle θ, red right-angle square, and sides labeled a, b, and c. Four graphs display the curves of sine, cosine, tangent, and cotangent functions, each labeled with x- and y-axes, key angles in radians, and typical wave or asymptote patterns.
Quizzes
Equations written on blackboard
Numbers and Mathematics
Quick Facts
Key People:
Abū al-Wafāʾ
Britannica AI Icon
Adding us as a Preferred Source tells Google you trust our content, so our articles show up more often in your Google Search, Discover feed, and AI Overviews.
verifiedCite
While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions.
Select Citation Style
print Print
Please select which sections you would like to print:
Feedback
Corrections? Updates? Omissions? Let us know if you have suggestions to improve this article (requires login).
Thank you for your feedback

Our editors will review what you’ve submitted and determine whether to revise the article.

External Websites
Diagram of a right triangle labeled with sides a, b, and hypotenuse c, alongside formulas for the six trigonometric functions-sine, cosine, tangent, cosecant, secant, and cotangent-each defined by ratios of the triangle's sides.
Trigonometric functions Based on the definitions, various simple relationships exist among the functions. For example, csc A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A, and tan A = sin A/cos A.

trigonometric function

mathematics
Also known as: circular function
Written and fact-checked by
Britannica Editors
Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree....
Britannica Editors
Last updated
•History
Top Questions

What are the six trigonometric functions?

How are trigonometric functions related to a unit circle?

What is the fundamental trigonometric identity?

How can trigonometric functions be applied in real-world problems?

What is the relationship between tangent and cotangent functions?

trigonometric function, in mathematics, one of the six functions (sine [sin], cosine [cos], tangent [tan], cotangent [cot], secant [sec], and cosecant [csc]) that represent ratios of sides of right triangles. These six functions form the foundation of trigonometry. They are also called the circular functions, since their values can be defined as ratios of the x and y coordinates (see coordinate system) of points on a circle of radius 1 (unit circle) that correspond to angles in standard positions. Notably, in a unit circle, the x coordinate of a point on the unit circle is equal to the cosine of the angle formed by the point, the origin, and the x-axis. Similarly, the y coordinate corresponds to the sine function for that angle.

Trigonometry can be easily applied to surveying, engineering, and navigation problems in which one of a right triangle’s acute angles and the length of a side are known and the lengths of the other sides are to be found. The fundamental trigonometric identity is sin2θ + cos2θ = 1, in which θ is an angle. Certain intrinsic qualities of the trigonometric functions make them useful in mathematical analysis. In particular, their derivatives form patterns that are useful for solving differential equations. For more information about trigonometric functions, see trigonometry.

The ratio method

Diagram of a right triangle with a yellow interior, green outline, green angle θ, red right-angle square, and sides labeled a, b, and c.
Right triangle diagram with labeled sides and anglesIllustration of a right triangle showing the opposite side (or perpendicular) a, the adjacent side (or base) b, and the hypoteneuse c, the right angle in red, and angle θ in green with a yellow background.

The simplest approach to understanding trigonometric functions is to view them in the context of a right-angled triangle, which is composed of three sides: the opposite side (denoted by a and also referred to as “perpendicular”), the adjacent side (denoted by b and also referred to as “base”), and the hypotenuse (denoted by c). The angle formed between the opposite and adjacent sides is 90°, and the angle between the hypotenuse and the adjacent side is θ. The six trigonometric functions in relation to the sides of a right-angled triangle are expressed as:

Equations written on blackboard
Britannica Quiz
Numbers and Mathematics
  • sin θ = opposite side/hypotenuse
  • cos θ = adjacent side/hypotenuse
  • tan θ = opposite side/adjacent side
  • csc θ = hypotenuse/opposite side
  • sec θ = hypotenuse/adjacent side
  • cot θ = adjacent side/opposite side

Based on those formulas, the following relations can be derived.

  • sin θ = 1/csc θ
  • cos θ = 1/sec θ
  • tan θ = 1/cot θ

Furthermore, the quotient relations can be established from the above-mentioned formulas.

  • tan θ = sin θ/cos θ
  • cot θ = cos θ/sin θ

Trigonometric identities

Trigonometric identities are derived from the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b); that is, a2 + b2 = c2.

When the hypotenuse has length 1, the Pythagorean theorem can be expressed as:sin2θ + cos2θ = 1

Explore Britannica Premium!

The trusted destination for professionals, college students, and lifelong learners.

SUBSCRIBE
Penguin, ship, mountain, atlas
shohei ohtani, plants, andy wharhol art
Mobile

From the definitions of the trigonometric functions, one can obtain the following identities.

  • 1 + tan2θ = sec2θ
  • 1 + cot2θ = cosec2θ
The Editors of Encyclopaedia BritannicaThis article was most recently revised and updated by Erik Gregersen.