In trigonometry, it is common to use mnemonics to help remember trigonometric identities and the relationships between the various trigonometric functions.
The sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters, for instance SOH-CAH-TOA in English:
One way to remember the letters is to sound them out phonetically (i.e. /ˌsoʊkəˈtoʊə/ SOH-kə-TOH-ə, similar to Krakatoa). [1]
Another method is to expand the letters into a sentence, such as "Some Old Horses Chew Apples Happily Throughout Old Age", "Some Old Hippy Caught Another Hippy Tripping On Acid", or "Studying Our Homework Can Always Help To Obtain Achievement". The order may be switched, as in "Tommy On A Ship Of His Caught A Herring" (tangent, sine, cosine) or "The Old Army Colonel And His Son Often Hiccup" (tangent, cosine, sine) or "Come And Have Some Oranges Help To Overcome Amnesia" (cosine, sine, tangent). [2] [3] Communities in Chinese circles may choose to remember it as TOA-CAH-SOH, which also means 'big-footed woman' (Chinese :大腳嫂; Pe̍h-ōe-jī :tōa-kha-só) in Hokkien.[ citation needed ]
An alternate way to remember the letters for Sin, Cos, and Tan is to memorize the syllables Oh, Ah, Oh-Ah (i.e. /oʊəˈoʊ.ə/ ) for O/H, A/H, O/A. [4] Longer mnemonics for these letters include "Oscar Has A Hold On Angie" and "Oscar Had A Heap of Apples." [2]
All Students Take Calculus is a mnemonic for the sign of each trigonometric functions in each quadrant of the plane. The letters ASTC signify which of the trigonometric functions are positive, starting in the top right 1st quadrant and moving counterclockwise through quadrants 2 to 4.
Other mnemonics include:
Other easy-to-remember mnemonics are the ACTS and CAST laws. These have the disadvantages of not going sequentially from quadrants 1 to 4 and not reinforcing the numbering convention of the quadrants.
Sines and cosines of common angles 0°, 30°, 45°, 60° and 90° follow the pattern with n = 0, 1, ..., 4 for sine and n = 4, 3, ..., 0 for cosine, respectively: [8]
0° = 0 radians | |||
30° = π/6 radians | |||
45° = π/4 radians | |||
60° = π/3 radians | |||
90° = π/2 radians | undefined |
Another mnemonic permits all of the basic identities to be read off quickly. The hexagonal chart can be constructed with a little thought: [9]
Starting at any vertex of the resulting hexagon:
Aside from the last bullet, the specific values for each identity are summarized in this table:
Starting function | ... equals 1/opposite | ... equals first/second clockwise | ... equals first/second counter-clockwise/anticlockwise | ... equals the product of two nearest neighbors |
---|---|---|---|---|
In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. The triangle's interior is a two-dimensional region. Sometimes an arbitrary edge is chosen to be the base, in which case the opposite vertex is called the apex.
In trigonometry, the law of tangents or tangent rule is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are great circles. Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and navigation.
In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half an angle is the stereographic projection of the circle through the point at angle radians onto the line through the angles . Among these formulas are the following:
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.
In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three points called angles or vertices.
The small-angle approximations can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians:
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle , the sine and cosine functions are denoted simply as and .
The following outline is provided as an overview of and topical guide to trigonometry:
There are several equivalent ways for defining trigonometric functions, and the proofs of the trigonometric identities between them depend on the chosen definition. The oldest and most elementary definitions are based on the geometry of right triangles. The proofs given in this article use these definitions, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles, see Trigonometric functions.
The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. For example, the derivative of the sine function is written sin′(a) = cos(a), meaning that the rate of change of sin(x) at a particular angle x = a is given by the cosine of that angle.
In mathematics, the values of the trigonometric functions can be expressed approximately, as in , or exactly, as in . While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots. The angles with trigonometric values that are expressible in this way are exactly those that can be constructed with a compass and straight edge, and the values are called constructible numbers.
Trigonometry is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios such as sine.
In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides and opposite respective angles and , the law of cosines states:
In trigonometry, Mollweide's formula is a pair of relationships between sides and angles in a triangle.
Solution of triangles is the main trigonometric problem of finding the characteristics of a triangle, when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as S1 because it is a one-dimensional unit n-sphere.
In mathematics, Niven's theorem, named after Ivan Niven, states that the only rational values of θ in the interval 0° ≤ θ ≤ 90° for which the sine of θ degrees is also a rational number are: