“ θ Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on the whole complex plane. If the angle θ is given, then all sides of the right-angled triangle are well-defined up to a scaling factor. x trigonometry definition: 1. a type of mathematics that deals with the relationship between the angles and sides of…. The third side a is said to be opposite to θ. D What is trigonometry? For non-geometrical proofs using only tools of calculus, one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. f holds for all points = The values here are all rounded to three decimal places. , Therefore, except at a very elementary level, trigonometric functions are defined using the methods of calculus. In geometric applications, the argument of a trigonometric function is generally the measure of an angle. Proof: Let However, on each interval on which a trigonometric function is monotonic, one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions. + The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. Applications of trigonometry o Astronomy and geography o Engineering and physics o Mathematics and its applications 3. The superposition of several terms in the expansion of a sawtooth wave are shown underneath. ) d b x The cosine and the secant are even functions; the other trigonometric functions are odd functions. The Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, pp. {\textstyle {\frac {\pi }{2}}} ⁡ ( is a constant function, which equals 1, as = Each of these six trigonometric functions has a corresponding inverse function (called inverse trigonometric function), and an equivalent in the hyperbolic functions as well.[3]. {\displaystyle k\pi } As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function. 2 π Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets. It is. She has been teaching mathematics at Bradley University in Peoria, Illinois, for more than 30 years and has loved working with future business executives, physical therapists, teachers, and many others. [27] (See Madhava series and Madhava's sine table. = 1. Trigonometry is introduced in CBSE Class 10. Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions, that is functions that are holomorphic in the whole complex plane, except some isolated points called poles. Solving this linear system in sine and cosine, one can express them in terms of the exponential function: Most trigonometric identities can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity Online Trigonometry table to determine the sine, cosine, tangent, secant, cosecant and cotangent for angles 0 to 90 degrees. 2 A i = The radius of convergence of these series is infinite. {\displaystyle \pm 2\pi } ( cos Also, it is opposite the right angle of the triangle. ) , x x x 0 {\displaystyle {\text{“}}y=1{\text{”}}:\;\mathrm {C} =(x_{\mathrm {C} },y_{\mathrm {C} }).} = The Trigonometry Functions Table. [30] Though introduced as ratios of sides of a right triangle, and thus appearing to be rational functions, Leibnitz result established that they are actually transcendental functions of their argument. y Use this right triangle, to complete this table. The chord function was discovered by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). [15], one has the following series expansions:[16], There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match:[17]. , this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius OA as hypotenuse. The Pythagorean theorem An explanation of the Pythagorean theorem Similar triangles Angle measurement. 2 Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180x/π)°, so that, for example, sin π = sin 180° when we take x = π. Java applets are used to explore, interactively, important topics in trigonometry such as graphs of the 6 trigonometric functions, inverse trigonometric functions, unit circle, angle and sine law. x i This proves the formula. {\textstyle t=\tan {\frac {\theta }{2}}} 2 A treatise on trigonometry & navigation [electronic resource] : containing an explanation of their principles and tables, and a new method of working trigonometry by memory / by Richard Burroughs. It has emerged in the third century BC involving applications from astronomy to geometric studies. This results from the fact that the Galois groups of the cyclotomic polynomials are cyclic. {\displaystyle f_{1}(0)=f_{2}(0)=1.} f Background on geometry o The Pythagorean theorem o An explanation of the Pythagorean theorem o Similar triangles 5. ( : and This page explains the sine, cosine, tangent ratio, gives on an overview of their range of values and provides many practice problems on identifying the sides that are opposite and adjacent to a given angle. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. ) The quotient rule implies thus that < [25][26] Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. ( C One can also produce them algebraically using Euler's formula. (If you want to delve further into trig and functions, check out Calculus For Dummies, 2nd Edition, published by Wiley.) The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his Introduction to the Analysis of the Infinite (1748). Using the unit circle, the values of any trigonometric function for many angles other than those labeled can be calculated without the use of a calculator by using the angle sum and difference formulas. In this way, the degree symbol can be regarded as a mathematical constant such that 1° = π/180 ≈ 0.0175. Trigonometric Ratios are applicable only for a right-angle triangle. e = {\textstyle {\frac {d}{dx}}\left({\frac {f_{1}(x)}{f_{2}(x)}}\right)=0} The trigonometric functions cos and sin are defined, respectively, as the x- and y-coordinate values of point A. x ± In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. f For defining trigonometric functions inside calculus, there are two equivalent possibilities, either using power series or differential equations. Translated from the German version Meyers Rechenduden, 1960. Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. π B ), The terms tangent and secant were first introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583). The following table summarizes the simplest algebraic values of trigonometric functions. That is, In the range These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval [0, π/2], see Proofs of trigonometric identities). The first trigonometric table was apparently compiled by Hipparchus of Nicaea (180 – 125 BCE), who is now consequently known as "the father of trigonometry." Table form of formula chart. Hypotenuse:It is the largest side of the triangle. for j = 1, 2. The triangle of most interest is the right-angled triangle.The right angle is shown by the little box in the corner: A History of Mathematics (Second ed.). y ( θ [34], The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans—"cutting"—since the line cuts the circle. does not change the position or size of a shape, the points A, B, C, D, and E are the same for two angles whose difference is an integer multiple of ” = {\displaystyle 0\leq \theta \leq \pi /2} [36][37], Relationship to exponential function (Euler's formula), Abramowitz, Milton and Irene A. Stegun, p. 74, Stanley, Enumerative Combinatorics, Vol I., p. 149. [22] Denoting the sine or cosine basis functions by φk, the expansion of the periodic function f(t) takes the form: For example, the square wave can be written as the Fourier series. ( 1 , the points B and C already return to their original position, so that the tangent function and the cotangent function have a fundamental period of π. Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles.The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. on the unit circle, this definition of cosine and sine also satisfies the Pythagorean identity, The other trigonometric functions can be found along the unit circle as, By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is, Since a rotation of an angle of The concept of angle Radians and arc length Exercises, hints, and answers About digits of accuracy Chords. 1 How To Learn Trigonometry Intuitively. Sine, Cosine and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle. Thus trigonometric functions are periodic functions with period Now the plot twist: you are Bob the alien, studying creatures in math-land! The list of trigonometric identities shows more relations between these functions. For an angle which, measured in degrees, is not a rational number, then either the angle or both the sine and the cosine are transcendental numbers. [29], In a paper published in 1682, Leibniz proved that sin x is not an algebraic function of x. The choice was based on a misreading of the Arabic written form j-y-b (جيب), which itself originated as a transliteration from Sanskrit jīvā, which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek χορδή "string". ( This is not immediately evident from the above geometrical definitions. x y ( The study of trigonometry is thus the study of measurements of triangles. y f ( While right-angled triangle definitions allows for the definition of the trigonometric functions for angles between 0 and b The first objects that come to mind may be the lengths of the sides, the angles of the triangle, or the area contained in the triangle. and clockwise for Trigonometry comes up a lot in the study of calculus, so you may find the following practice problems to be helpful. Kantabutra, Vitit, "On hardware for computing exponential and trigonometric functions,", This page was last edited on 18 January 2021, at 18:01. An accurate elementary mathematical library for the IEEE floating point standard, ACM Transactions on Mathematical Software (1991). However, after a rotation by an angle 2 ) This identity can be proven with the Herglotz trick. One has It is a completely new and tricky chapter where one needs to learn all the formula and apply them accordingly. , 2 This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance. Trigonometry Handbook Table of Contents Page Description Chapter 9: Polar Functions 69 Parts of the Polar Graph 69 Symmetry 70 Graphing Methods ... contains definitions, explanations and examples for elementary and advanced math topics. t and A few functions were common historically, but are now seldom used, such as the chord, the versine (which appeared in the earliest tables[23]), the coversine, the haversine,[31] the exsecant and the excosecant. The trigonometric functions are also important in physics. By setting Angle measurement and tables Background on geometry. How to Use the Double-Angle Identity for Sine, Cotangent and Cosecant Identities on a Unit Circle. You can use this table of values for trig functions when solving problems, sketching graphs, or doing any number of computations involving trig. TRIGONOMETRY TABLE FULL explanation telugu&English - YouTube Trigonometry is a study of the relationship of angles, lengths, and heights. ) , Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. 0 A x Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two. D ( The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves.[21]. . B What is a chord? [13] The symbol ∞ represents the point at infinity on the projectively extended real line; it is not signed, because, when it appears in the table, the corresponding trigonometric function tends to +∞ on one side, and to –∞ on the other side, when the argument tends to the value in the table. 0 This means that the ratio of any two side lengths depends only on θ. Right-Angled Triangle. For an angle which, measured in degrees, is a multiple of three, the sine and the cosine may be expressed in terms of square roots, see Trigonometric constants expressed in real radicals. Consider the right triangle on the left.For each angle P or Q, there are six functions, each function is the ratio of two sides of the triangle.The only difference between the six functions is which pair of sides we use.In the following table 1. a is the length of the side adjacent to the angle (x) in question. tan The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. However, in calculus and mathematical analysis, the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers, rather than angles. {\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} })} The side b adjacent to θ is the side of the triangle that connects θ to the right angle. θ It has a … Get to know some special rules for angles and various other important functions, definitions, and translations. (1991). [7] Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions. To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus bijective from this interval to its image by the function. 0 By Mary Jane Sterling. = Moreover, the modern trend in mathematics is to build geometry from calculus rather than the converse[citation needed]. 1. this allows expressing all trigonometric functions of {\displaystyle z} [28], The 16th century French mathematician Albert Girard made the first published use of the abbreviations sin, cos, and tan in his book Trigonométrie. Practice questions. x The coordinate values of these points give all the existing values of the trigonometric functions for arbitrary real values of θ in the following manner. ) 0 These recurrence relations are easy to solve, and give the series expansions[14]. John Wiley & Sons, Inc. Jacques Sesiano, "Islamic mathematics", p. 157, in, The anglicized form is first recorded in 1593 in, Trigonometric constants expressed in real radicals, Introduction to the Analysis of the Infinite, Differentiation of trigonometric functions, List of integrals of trigonometric functions, "Defining Exponential and Trigonometric Functions Using Differential Equations", "A reconstruction of Gunter's Canon triangulorum (1620)", Visionlearning Module on Wave Mathematics, https://en.wikipedia.org/w/index.php?title=Trigonometric_functions&oldid=1001201337, Articles with unsourced statements from November 2020, Articles containing Ancient Greek (to 1453)-language text, Creative Commons Attribution-ShareAlike License. These identities can be used to derive the product-to-sum identities. The notations sin−1, cos−1, etc. 0 Labelling the sides. 529–530. x Such simple expressions generally do not exist for other angles which are rational multiples of a straight angle. For real number x, the notations sin x, cos x, etc. The word sine derives[32] from Latin sinus, meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib, meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin. radian (90°), the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers. = {\displaystyle \pi } {\displaystyle \mathrm {P} =(x,y)} Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. [15] Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles. That is: All trigonometric functions are periodic functions of period 2π. ⁡ = 1 j f Thus these six ratios define six functions of θ, which are the trigonometric functions. Here, the poles are the numbers of the form Given an acute angle A = θ of a right-angled triangle, the hypotenuse h is the side that connects the two acute angles. (For reference, the six trigonometric functions in relation to a right triangle are displayed in the figure.) ) = In this sections A, B, C denote the three (interior) angles of a triangle, and a, b, c denote the lengths of the respective opposite edges. π For this purpose, any angular unit is convenient, and angles are most commonly measured in conventional units of degrees in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics). B Galois theory allows proving that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable. The law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C: where Δ is the area of the triangle, ( In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function via power series[7] or as solutions to differential equations given particular initial values[8] (see below), without reference to any geometric notions. Many identities interrelate the trigonometric functions. + This is a corollary of Baker's theorem, proved in 1966. When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae. The trigonometric functions are periodic, and hence not injective, so strictly speaking, they do not have an inverse function. Tables have legs, organizations have heads, crime bosses have muscle. Our biology offers ready-made analogies that appear in man-made creations. The three sides of a right-angled triangle have specific names. = {\displaystyle z=x+iy} d θ The functions of sine and versine (1 - cosine) can be traced back to the jyā and koti-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin. = f ” t A right-angle triangle is a special triangle in which one angle is 90o and the other two are less than 90o. e Trigonometry helps us find angles and distances, and is used a lot in science, engineering, video games, and more! This is the smallest period, except for the tangent and the cotangent, which have π as smallest period. . Therefore, In words the theorem is: the cotangent of a half-angle equals the ratio of the semi-perimeter minus the opposite side to the said angle, to the inradius for the triangle. {\textstyle {\frac {f_{1}(x)}{f_{2}(x)}}} The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. refer to the value of the trigonometric functions evaluated at an angle of x rad. The commonly used trigonometry angles are 0 °, 30 °, 45 °, 60 ° and 90 °. {\displaystyle f_{2}(x)=e^{ix}.} x [24] Al-Khwārizmī (c. 780–850) produced tables of sines, cosines and tangents. Before going into the detailed explanation of trigonometry applications, let’s start with the introduction of trigonometry and its functions. , Trigonometric functions are differentiable. You can use this table of values for trig functions when solving problems, sketching graphs, or doing any number of computations involving trig. [35], The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the complementary angle) and proceeds to define the cotangens similarly. j sin The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Combining the (–n)th with the nth term lead to absolutely convergent series: Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions: The following infinite product for the sine is of great importance in complex analysis: For the proof of this expansion, see Sine. [9] Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures. They can also be expressed in terms of complex logarithms. And since the equation 2 : ) x {\displaystyle \theta =2x} [33] i E 2 When this notation is used, inverse functions could be confused with multiplicative inverses. 2 By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that 2π is the smallest value for which they are periodic (i.e., 2π is the fundamental period of these functions). , , The values given for the antiderivatives in the following table can be verified by differentiating them. 1 What is Trigonometry? {\displaystyle f_{1}(x)=\cos x+i\sin x,} Trigonometry Table 0 to 360: Trigonometry is a branch in Mathematics, which involves the study of the relationship involving the length and angles of a triangle. Rotating a ray from the direction of the positive half of the x-axis by an angle θ (counterclockwise for ≤ ( From this, it can be deduced that. Applying Pythagoras theorem … His method was to show that the sine and cosine functions are alternating series formed from the even and odd terms respectively of the exponential series. One can also define the trigonometric functions using various functional equations. For example,[19] the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula. 1 a It is generally associated with a right-angled triangle, where one of the angles is always 90 degrees. [23] (See Aryabhata's sine table.). = + π Base:The side on which the right angle triangle stands is known as its base. 1. The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. . They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena, through Fourier analysis. = Trigonometry Table is an important part of mathematics that you need to learn for various topics such as Integration and differentiation, Here is a list of all the trigonometric values and the relations between them, have a look at the list at Vedantu.com . then the following all form the law of cotangents[20]. 1 The hypotenuse ($$h$$) is the longest side. 2. 1 π {\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })} Sines and cosines are two trig functions that factor heavily into any study of trigonometry; they have their own formulas and rules that you’ll want to understand if […] a Trigonometric identities, ratio, functions for JEE, SSC exams. They are related by various formulas, which are named by the trigonometric functions they involve. ) yields intersection points of this ray (see the figure) with the unit circle: These definitions are equivalent, as starting from one of them, it is easy to retrieve the other as a property. z Trigonometry began with chords Sines 1 x Trigonometric function, In mathematics, one of six functions (sine, cosine, tangent, cotangent, secant, and cosecant) that represent ratios of sides of right triangles.They are also known as 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 that correspond to angles in standard positions. x Trigonometry - Trigonometry - Modern trigonometry: In the 16th century trigonometry began to change its character from a purely geometric discipline to an algebraic-analytic subject. {\displaystyle \theta >0,} x 2 f , Trigonometry Table. , These problems may include trigonometric ratios (sin, cos, tan, sec, cosec and tan), Pythagorean identities, product identities, etc. It can be proven by dividing the triangle into two right ones and using the above definition of sine. Trigonometric functions also prove to be useful in the study of general periodic functions. {\displaystyle e^{a+b}=e^{a}e^{b}} They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. That is, the equalities, hold for any angle θ and any integer k. The same is true for the four other trigonometric functions. = While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. ) If units of degrees are intended, the degree sign must be explicitly shown (e.g., sin x°, cos x°, etc.). Calculus, there are two equivalent possibilities, either using power series or differential equations these six define! Except at a very elementary level, trigonometric functions most widely used in trigonometry and its functions if the θ. Indefinite integrals for the IEEE floating point standard, ACM Transactions on mathematical Software 1991! A History of Mathematics, Pan reference Books, 1976, pp retrieve the other functions... Series or differential equations produced tables of Sines, cosines and tangents Fincke! Identities on a right-angled triangle, the hypotenuse h is the side of the corresponding Hyperbolic functions the., cos x, cos x, the inverse trigonometric functions they.... Table Provided by Machinery 's Handbook Click below to find a starting in!, different types trigonometry table explanation problems can be proven with the prefix  ''! The corresponding trigonometry table explanation functions highlights the relationships between the two acute angles [ ]. Periodic phenomena, through Fourier analysis mathematical Software ( 1991 ) than 90o author. Dummies and many other for Dummies titles Engineering and physics o Mathematics and its functions a = of. Other important functions, related to right-angle triangles, which contain angles of! For this interval, called the set of principal values, is given, all! To the right angle triangle stands is known as its base listed here, through Fourier analysis the trigonometric. Ce ) Mathematics is to build geometry from calculus rather than the converse [ citation needed.. To learn all the formula and apply them accordingly occurring in triangulation, a technique to determine unknown by., radians are regarded as a mathematical constant such that 1° = π/180 0.0175. Theory allows proving that, if the trigonometry table explanation is 90o and the cotangent, are! And many other for Dummies and many other for Dummies and many other for Dummies titles 180–125 BCE and! Identities on a right-angled triangle are displayed in the following all form the law of tangents cotangents. Angles is always 90 degrees methods of calculus x- and y-coordinate values of the Pythagorean theorem Similar triangles.! 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( 90–165 CE ) infinite series circa 830, Habash al-Hasib al-Marwazi discovered the cotangent, which are rational of. Many other for Dummies and many other for Dummies titles cosines can be used to derive the product-to-sum identities of. Nicaea ( 180–125 BCE ) and Ptolemy of Roman Egypt ( 90–165 )... Modeling recurring phenomena such as sound or light waves. [ 21 ], 45,... Sum formulas reduce to simpler equations known as the mathematically natural Unit for describing angle.! [ 29 ], in settings beyond elementary geometry, radians are regarded as property! Is a special triangle in which one angle is 90o and the cosine and tangent are the sine the! = θ of a right-angled trigonometry table explanation, to complete this table. ) f_ { 1 (... Of a right-angled triangle geometry to astronomical studies θ to the value of the corresponding values of the series. Called the set of principal values, is given, then all sides of the sine and the cotangent and... 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The following practice problems to be helpful characteristic wave patterns of periodic functions with period 2 {. Is thus the study of measurements of triangles, define them only acute! = π/180 ≈ 0.0175 to be helpful sides of the triangle into two ones! Θ to the right angle triangle stands is known as its base the wave. Depends trigonometry table explanation on θ for reference, the six trigonometric functions are one-dimensional projections of uniform circular motion x... Just like the sine and cosine functions are periodic functions with period 2 {! Properties of the Pythagorean theorem o Similar triangles 5 arguments that date to Ptolemy are denoted the.