Trigonometry Edition 1.0 7th March 2006 From Wikibooks, the open-content textbooks collection Note: current version of this book can be found at http://en.wikibooks.org/wiki/Trigonometry TRIGONOMETRY ........................................................................................................................................................1 AUTHORS ...................................................................................................................................................................3 PREREQUISITES AND BASICS .....................................................................................................................................4 INTRODUCTION ..........................................................................................................................................................4 IN SIMPLE TERMS .......................................................................................................................................................4 Simple introduction ..............................................................................................................................................4 Angle-values simplified ........................................................................................................................................5 RADIAN AND DEGREE MEASURE ...............................................................................................................................7 A Definition and Terminology of Angles..............................................................................................................7 The radian measure .............................................................................................................................................7 Converting from Radians to Degrees...................................................................................................................9 Exercises ..............................................................................................................................................................9 THE UNIT CIRCLE ....................................................................................................................................................11 TRIGONOMETRIC ANGULAR FUNCTIONS..................................................................................................................12 Geometrically defining sin and cosine ...............................................................................................................12 Geometrically defining tangent..........................................................................................................................13 Domain and range of circular functions ............................................................................................................14 Applying the trigonometric functions to a right-angled triangle .......................................................................14 RIGHT ANGLE TRIGONOMETRY ...............................................................................................................................14 GRAPHS OF SINE AND COSINE FUNCTIONS ..............................................................................................................16 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS ....................................................................................................17 INVERSE TRIGONOMETRIC FUNCTIONS ....................................................................................................................21 The Inverse Functions, Restrictions, and Notation ............................................................................................21 The Inverse Relations.........................................................................................................................................21 APPLICATIONS AND MODELS ...................................................................................................................................23 Simple harmonic motion ....................................................................................................................................23 ANALYTIC TRIGONOMETRY .....................................................................................................................................26 USING FUNDAMENTAL IDENTITIES ..........................................................................................................................26 SOLVING TRIGONOMETRIC EQUATIONS ...................................................................................................................28 Basic trigonometric equations ...........................................................................................................................28 Further examples ...............................................................................................................................................33 SUM AND DIFFERENCE FORMULAS ..........................................................................................................................35 Cosine Formulas ................................................................................................................................................35 Sine Formulas ....................................................................................................................................................35 Tangent Formulas ..............................................................................................................................................35 Derivations.........................................................................................................................................................35 MULTIPLE-ANGLE AND PRODUCT-TO-SUM FORMULAS ...........................................................................................37 Multiple-Angle Formulas ...................................................................................................................................37 Proofs for Double Angle Formulas....................................................................................................................37 ADDITIONAL TOPICS IN TRIGONOMETRY .................................................................................................................38 LAW OF SINES..........................................................................................................................................................38 LAW OF COSINES .....................................................................................................................................................39 VECTORS AND DOT PRODUCTS ................................................................................................................................41 TRIGONOMETRIC FORM OF THE COMPLEX NUMBER ................................................................................................41 TRIGONOMETRY REFERENCES .................................................................................................................................42 TRIGONOMETRIC FORMULA REFERENCE .................................................................................................................42 TRIGONOMETRIC IDENTITIES REFERENCE................................................................................................................42 Pythagoras .........................................................................................................................................................42 Sum/Difference of angles ...................................................................................................................................42 Product to Sum...................................................................................................................................................43 Sum and difference to product ...........................................................................................................................43 Double angle......................................................................................................................................................43 2 Half angle...........................................................................................................................................................44 Power Reduction ................................................................................................................................................44 Even/Odd............................................................................................................................................................44 Calculus .............................................................................................................................................................45 NATURAL TRIGONOMETRIC FUNCTIONS OF PRIMARY ANGLES ...............................................................................46 LICENSE ...................................................................................................................................................................48 GNU Free Documentation License ....................................................................................................................48 0. PREAMBLE ...................................................................................................................................................48 1. APPLICABILITY AND DEFINITIONS ..........................................................................................................48 2. VERBATIM COPYING...................................................................................................................................49 3. COPYING IN QUANTITY..............................................................................................................................49 4. MODIFICATIONS .........................................................................................................................................49 5. COMBINING DOCUMENTS.........................................................................................................................50 6. COLLECTIONS OF DOCUMENTS ..............................................................................................................50 7. AGGREGATION WITH INDEPENDENT WORKS .......................................................................................51 8. TRANSLATION ..............................................................................................................................................51 9. TERMINATION..............................................................................................................................................51 10. FUTURE REVISIONS OF THIS LICENSE..................................................................................................51 External links .....................................................................................................................................................51 Authors Lmov, Evil Saltine, JEdwards, llg, Programmermatt, Alsocal (Wikibooks Users) 3 Prerequisites And Basics To be able to study Trigonometry sucessfully, it is recommended that students complete; Geometry, Algebra I and Algebra II prior to digging in to the course material. It is helpful to have a graphing calculator and graph paper on hand to be able to follow along as well. If one is not available software available on sites such as http://www.graphcalc.com/ may be helpful. Introduction Trigonometry is an important, fundamental step in math education. From the seemingly simple shape, the right triangle, we gain tools and insight that help us in further practical as well as theoretical endeavors. The subtle mathematical relationships between the right triangle, the circle, the sine wave, and the exponential curve can only be fully understood with a firm basis in trigonometry. In simple terms This page is intended as a simplified introduction to trigonometry. (This article is not always correctly formulated in mathematical language.) Simple introduction If you are unfamiliar with angles, where they come from, and why they are actually required, this section will help you develop your understanding. In principle, all angles and trigonometric functions are defined on the unit circle. The term unit in mathematics applies to a single measure of any length. We will later apply the principles gleaned from unit measures to a larger (or smaller) scaled problems. All the functions we need can be derived from a triangle inscribed in the unit circle: it happens to be a right-angled triangle. A Right Triangle 4 The center point of the unit circle will be set on a Cartesian plane, with the circle's centre at the origin of the plane — the point (0,0). Thus our circle will be divided into four sections, or quadrants. Quadrants are always counted counter-clockwise, as is the default rotation of angular velocity ω (omega). Now we inscribe a triangle in the first quadrant (that is, where the x- and y-axes are assigned positive values) and let one leg of the angle be on the x-axis and the other be parallel to the y-axis. (Just look at the illustration for clarification). Now we let the hypotenuse (which is always 1, the radius of our unit circle) rotate counter-clockwise. You will notice that a new triangle is formed as we move into a new quadrant, not only because the sum of a triangle's angles cannot be beyond 180°, but also because there is no way on a two-dimensional plane to imagine otherwise. Angle-values simplified Imagine the angle to be nothing more than exactly the size of the triangle leg that resides on the x-axis (the cosine). So for any given triangle inscribed in the unit circle we would have an angle whose value is the distance of the triangle-leg on the x-axis. Although this would be possible in principle, it is much nicer to have a independent variable, let's call it phi, which does not change sign during the change from one quadrant into another and is easier to handle (that means it is not necessarily always a decimal number). !!Notice that all sizes and therefore angles in the triangle are mutually directly proportional. So for instance if the x-leg of the triangle is short the y-leg gets long. That is all nice and well, but how do we get the actual length then of the various legs of the triangle? By using translation tables, represented by a function (therefore arbitrary interpolation is possible) that can be composed by algorithms such as taylor. Those translation-table-functions (sometimes referred to as LUT, Look up tables) are well known to everyone and are known as sine, cosine and so on. (Whereas of course all the abovementioned latter ones can easily be calculated by using the sine and cosine). In fact in history when there weren't such nifty calculators available, printed sine and cosine tables had to be used, and for those who needed interpolated data of arbitrary accuracy - taylor was the choice of word. So how can I apply my knowledge now to a circle of any scale. Just multiply the scaling coefficient with the result of the trigonometric function (which is referring to the unit circle). And this is also why cos(φ)2 + sin(φ)2 = 1, which is really nothing more than a veiled version of the pythagorean theorem: cos(φ) = a;sin(φ) = b;a2 + b2 = c2, whereas the c = 12 = 1, a peculiarity of most unit constructs. Now you also see why it is so comfortable to use all those mathematical unit-circles. Another way to interprete a angle-value would be: A angle is nothing more than a translated 'directed'-length into which the information of the actual quadrant is packed and the applied type of trigonometric function along with its sign determines the axis ('direction'). Thus something 5 like the translation of a (x,y)-tuple into polar coordinates is a piece of cake. However due to the fact that information such as the actual quadrant is 'translated' from the sign of x and y into the angular value (a multitude of 90) calculations such as for instance the division in polar-form isn't equal to the steps taken in the non-polar form. Oh and watch out to set the right signs in regard to the number of quadrant in which your triangle is located. (But you'll figure that out easily by yourself). I hope the magic behind angles and trigonometric functions has disappeared entirely by now, and will let you enjoy a more in-depth study with the text underneath as your personal tutor. 6 Radian and Degree Measure A Definition and Terminology of Angles An angle is determined by rotating a ray about its endpoint. The starting position of the ray is called the initial side of the angle. The ending position of the ray is called the terminal side. The endpoint of the ray is called its vertex. Positive angles are generated by counter-clockwise rotation. Negative angles are generated by clockwise rotation. Consequently an angle has four parts: its vertex, its initial side, its terminal side, and its rotation. An angle is said to be in standard position when it is drawn in a cartesian coordinate system in such a way that its vertex is at the origin and its initial side is the positive x-axis. The radian measure One way to measure angles is in radians. To signify that a given angle is in radians, a superscript c, or the abbreviation rad might be used. If no unit is given on an angle measure, the angle is assumed to be in radians. Defining a radian A single radian is defined as the angle formed in the minor sector of a circle, where the minor arc length is the same as the radius of the circle. 7 Defining a radian with respect to the unit circle. Measuring an angle in radians The size of an angle, in radians, is the length of the circle arc s divided by the circle radius r. (rad). Because we know the circumference of a circle to be equal to 2πr, it follows that a central angle of one full counterclockwise revolution gives an arc length (or circumference) of s = 2πr. Thus 2 8 π radians corresponds to 360°, that is, there are 2π radians in a circle. Converting from Radians to Degrees Because there are 2π radians in a circle: To convert degrees to radians: (rad) To convert radians to degrees: Exercises Excercise 1 Convert the following angle measurements from degrees to radians. Express your answer exactly (in terms of π). a) 180 degrees b) 90 degrees c) 45 degrees d) 137 degrees Exercise 2 Convert the following angle measurements from radians to degrees. 5. 6. 7. 9 8. Answers Exercise 1 a) π b) c) d) Exercise 2 a) 60° b) 30° c) 15° d) 135° 10 The Unit Circle The Unit Circle is a circle with its center at the origin (0,0) and a radius of one unit. The Unit Circle Angles are always measured from the positive x-axis (also called the "right horizon"). Angles measured counterclockwise have positive values; angles measured clockwise have negative values. A unit circle with certain exact values marked on it is available at Wikipedia. 11 Trigonometric Angular Functions Geometrically defining sin and cosine In the unit circle shown here, a unit-length radius has been drawn from the origin to a point (x,y) on the circle. Defining sine and cosine A line perpendicular to the x-axis, drawn through the point (x,y), intersects the x-axis at the point with the abscissa x. Similarly, a perpendicular to the y-axis intersects the y-axis at the point with the ordinate y. The angle between the x-axis and the radius is α. We define the basic trigonometric functions of any angle α as follows: 12 tanθ can be algebraically defined. These three trigonometric functions can be used whether the angle is measured in degrees or radians as long as it specified which, when calculating trigonometric functions from angles or vice versa. Geometrically defining tangent In the previous section, we algebraically defined tangent, and this is the definition that we will use most in the future. It can, however, be helpful to understand the tangent function from a geometric perspective. Geometrically defining tangent A line is drawn at a tangent to the circle: x = 1. Another line is drawn from the point on the radius of the circle where the given angle falls, through the origin, to a point on the drawn tangent. The ordinate of this point is called the tangent of the angle. 13 Domain and range of circular functions Any size angle can be the input to sine or cosine — the result will be as if the largest multiple of 2π (or 360°) were subtracted from the angle. The output of the two functions is limited by the absolute value of the radius of the unit circle, | 1 | . R represents the set of all real numbers. No such restrictions apply to the tangent, however, as can be seen in the diagram in the preceding section. The only restriction on the domain of tangent is that odd multiples of undefined, as a line parallel to the tangent will never intersect it. are Applying the trigonometric functions to a right-angled triangle If you redefine the variables as follows to correspond to the sides of a right triangle: - x = a (adjacent) - y = o (opposite) - a = h (hypotenuse) Right Angle Trigonometry We have defined the sine, cosine, and tangent functions using the unit circle. Now we can apply them to a right triangle. 14 This triangle has sides A and B. The angle between them, c, is a right angle. The third side, C, is the hypotenuse. Side A is opposite angle a, and side B is adjacent to angle a. Applying the definitions of the functions, we arrive at these useful formulas: sin(a) = A / C cos(a) = B / C tan(a) = A / B or opposite side over hypotenuse or adjacent side over hypotenuse or opposite side over adjacent side Exercises: (Draw a diagram!) 1. A right triangle has side A = 3, B = 4, and C = 5. Calculate the following: sin(a), cos(a), tan(a) 2. A different right triangle has side C = 6 and sin(a) = 0.5 . Calculate side A. 15 Graphs of Sine and Cosine Functions The graph of the sine function looks like this: The graph of the cosine function looks like this: Sine and cosine are periodic functions; that is, the above is repeated for preceding and following intervals with length 2π. 16 Graphs of Other Trigonometric Functions A graph of tan(x). 17 A graph of sec(x). sec(x) is defined as . 18 A graph of csc(x). csc(x) is defined as . 19 A graph of cot(x). cot(x) is defined as or Note that tan(x), sec(x), and csc(x) are unbounded. 20 . Inverse Trigonometric Functions The Inverse Functions, Restrictions, and Notation While it might seem that inverse trigonometric functions should be relatively self defining, some caution is necessary to get an inverse function since the trigonometric functions are not one-to-one. To deal with this issue, some texts have adopted the convention of allowing sin − 1x, cos − 1x, and tan − 1x (all with lower-case initial letters) to indicate the inverse relations for the trigonometric functions and defining new functions Sinx, Cosx, and Tanx (all with initial capitals) to equal the original functions but with restricted domain, thus creating one-to-one functions with the inverses Sin − 1x, Cos − 1x, and Tan − 1x. For clarity, we will use this convention. Another common notation used for the inverse functions is the "arcfunction" notation: Sin − 1x = arcsinx, Cos − 1x = arccosx, and Tan − 1x = arctanx (the arcfunctions are sometimes also capitalized to distinguish the inverse functions from the inverse relations). The arcfunctions may be so named because of the relationship between radian measure of angles and arclength--the arcfunctions yeild arc lengths on a unit circle. The restrictions necessary to allow the inverses to be functions are standard: Sin − 1x has range ; Cos − 1x has range ; and Tan − 1x has range (these restricted ranges for the inverses are the restricted domains of the capital-letter trigonometric functions). For each inverse function, the restricted range includes first-quadrant angles as well as an adjacent quadrant that completes the domain of the inverse function and maintains the range as a single interval. It is important to note that because of the restricted ranges, the inverse trigonometric functions do not necessarily behave as one might expect an inverse function to behave. While (following the expected ), ! For the inverse trigonometric functions, only when x is in the range of the inverse function. The other direction, however, is less tricky: for all x to which we can apply the inverse function. The Inverse Relations For the sake of completeness, here are definitions of the inverse trigonometric relations based on the inverse trigonometric functions: • (the sine function has period 2π, but within any given period may have two solutions and ) 21 • (the cosine function has period 2π, but within any given period may have two solutions and cosine is even-) • (the tangent function has period π and is oneto-one within any given period) 22 Applications and Models Simple harmonic motion Simple harmonic motion. Notice that the position of the dot matches that of the sine wave. Simple harmonic motion (SHM) is the motion of an object which can be modeled by the following function: or where c1 = A sin Ï† and c2 = A cos Ï†. In the above functions, A is the amplitude of the motion, Ï‰ is the angular velocity, and Ï† is the phase. The velocity of an object in SHM is The acceleration is Springs and Hooke's Law An application of this is the motion of a weight hanging on a spring. The motion of a spring can be modeled approximately by Hooke's Law: F = -kx where F is the force the spring exerts, x is the position of the end of the spring, and k is a constant characterizing the spring (the stronger the spring, the higher the constant). 23 Calculus-based derivation From Newton's laws we know that F = ma where m is the mass of the weight, and a is its acceleration. Substituting this into Hooke's Law, we get ma = -kx Dividing through by m: The calculus definition of acceleration gives us Thus we have a second-order differential equation. Solving it gives us (2) with an independent variable t for time. We can change this equation into a simpler form. By lettting c1 and c2 be the legs of a right triangle, with angle Ï† adjacent to c2, we get and 24 Substituting into (2), we get Using a trigonometric identity, we get: (3) Let and . Substituting this into (3) gives 25 Analytic Trigonometry Using Fundamental Identities Some of the fundamental trigometric identities are those derived from the Pythagorean Theorem. These are defined using a right triangle: By the Pythagorean Theorem, A2 + B2 = C2 {1} Dividing through by C2 gives {2} We have already defined the sine of a in this case as A/C and the cosine of a as B/C. Thus we can substitute these into {2} to get sin2a + cos2a = 1 Related identities include: sin2a = 1 − cos2a or cos2a = 1 − sin2a tan2a + 1 = sec2a or tan2a = sec2a − 1 1 + cot2a = csc2a or cot2a = csc2a − 1 Other Fundamental Identities include the Reciprocal, Ratio, and Co-function identities 26 Reciprocal identities Ratio identities Co-function identities (in radians) 27 Solving Trigonometric Equations Trigonometric equations involve finding an unknown which is an argument to a trigonometric function. Basic trigonometric equations sin x = n |n|>1 28 The equation sinx = n has solutions only when n is within the interval [-1; 1]. If n is within this interval, then we need to find and α such that: The solutions are then: Where k is an integer. In the cases when n equals 1, 0 or -1 these solutions have simpler forms which are summarizied in the table on the right. For example, to solve: First find α: Then substitute in the formulae above: Solving these linear equations for x gives the final answer: Where k is an integer. cos x = n 29 |n|>1 Like the sine equation, an equation of the form cosx = n only has solutions when n is in the interval [-1; 1]. To solve such an equation we first find the angle α such that: Then the solutions for x are: Where k is an integer. Simpler cases with n equal to 1, 0 or -1 are summarized in the table on the right. tan x = n 30 General case An equation of the form tanx = n has solutions for any real n. To find them we must first find an angle α such that: After finding α, the solutions for x are: When n equals 1, 0 or -1 the solutions have simpler forms which are shown in the table on the right. cot x = n 31 General case The equation cotx = n has solutions for any real n. To find them we must first find an angle α such that: After finding α, the solutions for x are: When n equals 1, 0 or -1 the solutions have simpler forms which are shown in the table on the right. csc x = n and sec x = n The trigonometric equations csc x = n and sec x = n can be solved by transforming them to other basic equations: 32 Further examples Generally, to solve trigonometric equations we must first transform them to a basic trigonometric equation using the trigonometric identities. This sections lists some common examples. a sin x + b cos x = c To solve this equation we will use the identity: 0 \\ \pi + \tan^{-1} \left(b / a\right), && \mbox{if } a The equation becomes: This equation is of the form sinx = n and can be solved with the formulae given above. For example we will solve: In this case we have: Apply the identity: 33 So using the formulae for sinx = n the solutions to the equation are: 34 Sum and Difference Formulas Cosine Formulas cos(a + b) = cosacosb − sinasinb cos(a − b) = cosacosb + sinasinb Sine Formulas sin(a + b) = sin a cos b + cos a sin b sin(a - b) = sin a cos b - cos a sin b sin2a = 2 sin a cos a Tangent Formulas tan(a +b) = (tan a + tan b)/(1 - tan a tan b) tan(a - b) = (tan a -tan b)/(1 + tan a tan b) Derivations • cos(a + b) = cos a cos b - sin a sin b • cos(a - b) = cos a cos b + sin a sin b Using cos(a + b) and the fact that cosine is even and sine is odd, we have cos(a + (-b)) = cos a cos (-b) - sin a sin (-b) = cos a cos b - sin a (-sin b) = cos a cos b + sin a sin b • sin(a + b) = sin a cos b + cos a sin b Using cofunctions we know that sin a = cos (90 - a). Use the formula for cos(a - b) and cofunctions we can write sin(a + b) = = = = • cos(90 - (a + b)) cos((90 - a) - b) cos(90 -a)cos b + sin(90 - a)sin b sin a cos b + cos a sin b sin(a - b) = sin a cos b - cos a sin b 35 Having derived sin(a + b) we replace b with "-b" and use the fact that cosine is even and sine is odd. sin(a + (-b)) = sin a cos (-b) + cos a sin (-b) = sin a cos b + cos a (-sin b) = sin a cos b - cos a sin b 36 Multiple-Angle and Product-to-sum Formulas Multiple-Angle Formulas • sin(2a) = 2 sin a cos a • cos(2a) = cos2 a - sin2 a = 1 - 2 sin2 a = 2 cos2 a - 1 • tan(2a) = (2 tan a)/(1 - tan2 a) Proofs for Double Angle Formulas • cos(a + b) = cos(a)cos(b) − sin(a)sin(b) ; a = b for cos(2a) cos (a) − sin2(a) = cos(2a) cos2(a) − (1 − cos2(a)) = 2cos22(a) − 1 = cos(2a) (Note that sin2(x) = 1 − cos2(x) 2 • sin(a + b) = sin(a)cos(b) + sin(b)cos(a) ; a = b for sin(2a) sin(a)cos(a) + sin(a)cos(a) = 2cos(a)sin(a) = sin(2a) Trigonometry 37 Additional Topics in Trigonometry Law of Sines Consider this triangle: It has three sides • A, length A, opposite angle a at vertex a • B, length B, opposite angle b at vertex b • C, length C, opposite angle c at vertex c The perpendicular, oc, from line ab to vertex c has length h The Law of Sines states that: The law can also be written as the reciprocal: 38 Proof The perpendicular, oc, splits this triangle into two right-angled triangles. This lets us calculate h in two different ways • Using the triangle cao gives • Using the triangle cbo gives • Eliminate h from these two equations • Rearrange By using the other two perpendiculars the full law of sines can be proved. Law of Cosines Consider this triangle: 39 It has three sides • a, length a, opposite angle A at vertex A • b, length b, opposite angle B at vertex B • c, length c, opposite angle C at vertex C The perpendicular, oc, from line ab to vertex c has length h The Law Of Cosines states that: Proof The perpendicular, oc, divides this triangle into two right angled triangles, aco and bco. First we will find the length of the other two sides of triangle aco in terms of known quantities, using triangle bco. h=a sin B Side c is split into two segments, total length c. ob, length c cos B ao, length c - a cos B Now we can use Pythagoras to find b, since b2 = "ao"2 + h2 The corresponding expressions for a and c can be proved similarly. 40 Vectors and Dot Products Consider the vectors U and V (with respective magnitudes |U| and |V|). If those vectors enclose an angle θ then the dot product of those vectors can be written as: If the vectors can be written as: then the dot product is given by: Trigonometric Form of the Complex Number where • i is the Imaginary Number • the modulus • the argument is the angle formed by the complex number on a polar graph with one real axis and one imaginary axis. This can be found using the right angle trigonometry for the trigonometric functions. This is sometimes abbreviated as and it is also the case that (provided that φ is in radians). 41 Trigonometry References Trigonometric Formula Reference The principal identity in trigonometry is: sin2θ + cos2θ = 1 All trigonometric function are 2π periodic: sinθ = sin(θ + 2π) cosθ = cos(θ + 2π) tanθ = tan(θ + 2π) cotθ = cos(θ + 2π) cscθ = cos(θ + 2π) secθ = cos(θ + 2π) Formulas involving sums of angles are as follows: sin(α + β) = sinαcosβ + cosαsinβ cos(α + β) = cosαcosβ − sinαsinβ Substituting α = β gives the double angle formulae Trigonometric Identities Reference Pythagoras 1. sin2(x) + cos2(x) = 1 2. 1 + tan2(x) = sec2(x) 3. 1 + cot2(x) = csc2(x) These are all direct consequences of Pythagoras's theorem. Sum/Difference of angles 1. 42 2. 3. Product to Sum 1. 2sin(x)sin(y) = cos(x - y) - cos(x + y) 2. 2cos(x)cos(y) = cos(x - y) + cos(x + y) 3. 2sin(x)cos(y) = sin(x - y) + sin(x + y) Sum and difference to product 1. Asin(x) + Bcos(x) = Csin(x + y) where 1. 2. 3. 4. Double angle 1. cos(2x) = cos2(x) − sin2(x) = 2cos2(x) − 1 = 1 − 2sin2(x) 2. sin(2x) = 2sin(x)cos(x) 3. These are all direct consequences of the sum/difference formulae 43 Half angle 1. 2. 3. In cases with , the sign of the result must be determined from the value of from the cos(2x) formulae. Power Reduction 1. 2. 3. Even/Odd 1. sin( − θ) = − sin(θ) 2. cos( − θ) = cos(θ) 3. tan( − θ) = − tan(θ) 4. csc( − θ) = − csc(θ) 5. sec( − θ) = sec(θ) 6. cot( − θ) = − cot(θ) 44 . These derive Calculus 1. 2. 3. 4. 5. 6. 45 Natural Trigonometric Functions of Primary Angles θ (radians) sinθ cosθ tanθ cotθ secθ cscθ 0 0 1 0 undef 1 undef 2 1 1 2 1 0 undef 0 undef 1 -2 -1 -1 2 π 0 -1 0 undef -1 undef -2 -1 -1 -2 -1 0 undef 0 undef -1 2 -1 -1 -2 2π 0 1 0 undef 46 1 undef θ (degrees) Note: some values in the table are given in forms that include a radical in the denominator--this is done both to simplify recognition of reciprocal pairs and because the form given in the table is simpler in some sense. 47 License GNU Free Documentation License Version 1.2, November 2002 Copyright (C) 2000,2001,2002 Free Software Foundation, Inc. 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. 0. 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Examples of suitable formats for Transparent copies include plain ASCII without markup, Texinfo input format, LaTeX input format, SGML or XML using a publicly available DTD, and standard-conforming simple HTML, PostScript or PDF designed for human modification. Examples of transparent image formats include PNG, XCF and JPG. Opaque formats include proprietary formats that can be read and edited only by proprietary word processors, SGML or XML for which the DTD and/or processing tools are not generally available, and the machine-generated HTML, PostScript or PDF produced by some word processors for output purposes only. The "Title Page" means, for a printed book, the title page itself, plus such following pages as are needed to hold, legibly, the material this License requires to appear in the title page. For works in formats which do not have any title page as such, "Title Page" means the text near the most prominent appearance of the work's title, preceding the beginning of the body of the text. 48 A section "Entitled XYZ" means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language. (Here XYZ stands for a specific section name mentioned below, such as "Acknowledgements", "Dedications", "Endorsements", or "History".) To "Preserve the Title" of such a section when you modify the Document means that it remains a section "Entitled XYZ" according to this definition. The Document may include Warranty Disclaimers next to the notice which states that this License applies to the Document. These Warranty Disclaimers are considered to be included by reference in this License, but only as regards disclaiming warranties: any other implication that these Warranty Disclaimers may have is void and has no effect on the meaning of this License. 2. VERBATIM COPYING You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License. You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute. However, you may accept compensation in exchange for copies. If you distribute a large enough number of copies you must also follow the conditions in section 3. You may also lend copies, under the same conditions stated above, and you may publicly display copies. 3. COPYING IN QUANTITY If you publish printed copies (or copies in media that commonly have printed covers) of the Document, numbering more than 100, and the Document's license notice requires Cover Texts, you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers must also clearly and legibly identify you as the publisher of these copies. The front cover must present the full title with all words of the title equally prominent and visible. You may add other material on the covers in addition. Copying with changes limited to the covers, as long as they preserve the title of the Document and satisfy these conditions, can be treated as verbatim copying in other respects. If the required texts for either cover are too voluminous to fit legibly, you should put the first ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent pages. If you publish or distribute Opaque copies of the Document numbering more than 100, you must either include a machine-readable Transparent copy along with each Opaque copy, or state in or with each Opaque copy a computer-network location from which the general network-using public has access to download using public-standard network protocols a complete Transparent copy of the Document, free of added material. If you use the latter option, you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy (directly or through your agents or retailers) of that edition to the public. It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document. 4. MODIFICATIONS You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version: A. Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document). You may use the same title as a previous version if the original publisher of that version gives permission. B. List on the Title Page, as authors, one or more persons or entities responsible for authorship of the modifications in the Modified Version, together with at least five of the principal authors of the Document (all of its principal authors, if it has fewer than five), unless they release you from this requirement. C. State on the Title page the name of the publisher of the Modified Version, as the publisher. D. Preserve all the copyright notices of the Document. E. Add an appropriate copyright notice for your modifications adjacent to the other copyright notices. F. Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the Addendum below. G. Preserve in that license notice the full lists of Invariant Sections and required Cover Texts given in the Document's license notice. 49 H. Include an unaltered copy of this License. I. Preserve the section Entitled "History", Preserve its Title, and add to it an item stating at least the title, year, new authors, and publisher of the Modified Version as given on the Title Page. If there is no section Entitled "History" in the Document, create one stating the title, year, authors, and publisher of the Document as given on its Title Page, then add an item describing the Modified Version as stated in the previous sentence. J. Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document for previous versions it was based on. These may be placed in the "History" section. You may omit a network location for a work that was published at least four years before the Document itself, or if the original publisher of the version it refers to gives permission. K. For any section Entitled "Acknowledgements" or "Dedications", Preserve the Title of the section, and preserve in the section all the substance and tone of each of the contributor acknowledgements and/or dedications given therein. L. Preserve all the Invariant Sections of the Document, unaltered in their text and in their titles. Section numbers or the equivalent are not considered part of the section titles. M. Delete any section Entitled "Endorsements". Such a section may not be included in the Modified Version. N. Do not retitle any existing section to be Entitled "Endorsements" or to conflict in title with any Invariant Section. O. Preserve any Warranty Disclaimers. If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version's license notice. These titles must be distinct from any other section titles. You may add a section Entitled "Endorsements", provided it contains nothing but endorsements of your Modified Version by various parties--for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard. You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one. The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version. 5. COMBINING DOCUMENTS You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers. The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work. In the combination, you must combine any sections Entitled "History" in the various original documents, forming one section Entitled "History"; likewise combine any sections Entitled "Acknowledgements", and any sections Entitled "Dedications". You must delete all sections Entitled "Endorsements." 6. COLLECTIONS OF DOCUMENTS You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects. You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document. 50 7. AGGREGATION WITH INDEPENDENT WORKS A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an "aggregate" if the copyright resulting from the compilation is not used to limit the legal rights of the compilation's users beyond what the individual works permit. When the Document is included in an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document. If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document's Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate. 8. TRANSLATION Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail. If a section in the Document is Entitled "Acknowledgements", "Dedications", or "History", the requirement (section 4) to Preserve its Title (section 1) will typically require changing the actual title. 9. TERMINATION You may not copy, modify, sublicense, or distribute the Document except as expressly provided for under this License. Any other attempt to copy, modify, sublicense or distribute the Document is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance. 10. FUTURE REVISIONS OF THIS LICENSE The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/. Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License "or any later version" applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation External links • GNU Free Documentation License (Wikipedia article on the license) • Official GNU FDL webpage 51

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