May 9, 200211. The Trigonometric Functions11.1 REVIEW OF THE TRIGONOMETRIC FUNCTIONSAngular variables are often denoted by Greek letters, such as θ or φ. Wewill use θ. The angle should be measured in radians. See Figurefig:radians??. If sis the arclength of a section of a circle with radius r, then the angle of thesection, in radians, is θ = s/r. For example, the angle of a semicircular arcis θ = πr/r = π radians, because the arclength of the half circle is πr. Thisangle is 180 degrees, so to convert between radians and degrees we use theequivalenceπ radians = 180 degrees.For example, a right angle is 90 degrees or π/2 radians.The well-known trigonometric functions are sin θ, cos θ, and tan θ. Theseare defined from the lengths in a right triangle in Figurefig:rightT??. Two importantidentities of the trig functions can be derived from the right-triangle formulae.First, note thatsin θcos θ=O/HA/H=OA= tan θ =⇒sin θcos θ= tan θ. (11-1)eq:id1Second, by the Pythagorean theorem,sin2θ + cos2θ =O2+ A2H2=H2H2= 1=⇒ sin2θ + cos2θ = 1. (11-2)eq:id2Less familiar trig functions are csc θ, sec θ, and cot θ, which are the reciprocalscsc θ =1sin θ, sec θ =1cos θ, cot θ =1tan θ. (11-3)We’ll analyze trig functions with θ as the independent variable. What arethe domain and range? The trigonometric relations in Fig.fig:rightT?? are limitedto 0 < θ < π/2, because the angles in a right triangle must be less than90 degrees. But we extend the definitions to angles greater than π/2 in Fig.fig:allangles??. Note that the sign of each trig function is negative for some range of θ.Figurefig:gt2pi?? shows the signs of the trig functions for angles greater than π/2.Because θ is an angle (in radians) it could be restricted to [0, 2π]. However,it is more useful to allow θ to take all real values, so that the domain is12 Chapter 11−∞ < θ < ∞. Because the angle θ + 2π corresponds to the same point P inthe plane as θ, all the trig functions are periodic functions; in particular,sin(θ + 2π) = sin θ, (11-4)cos(θ + 2π) = cos θ. (11-5)In graphical terms, a graph of sin θ or cos θ oscillates between +1 and −1.The curve repeats—taking exactly the same shape—over intervals of length2π. For example, cos θ goes from 1 to −1 and back to 1 as θ varies from 0to 2π; then it repeats with the same shape as θ varies from 2π to 4π; and itrepeats identically for any interval from 2πn to 2π(n + 1) with n an integer.The graphs of sin θ and cos θ are shown in Fig.fig:sincos??.1From the graphs wesee that the range of either sin θ or cos θ is [−1, 1]. Also, these functions arecontinuous.The function tan θ has separate branches. A graph of tan θ is shown inFig.fig:tan??.2The range of tan θ is (−∞, +∞). By the identity (eq:id111-1), tan θapproaches ±∞ as θ approaches any value with cos θ = 0. The right-triangleformulae (see Fig.fig:rightT??) show that cos(π/2) = 0, because the adjacent lengthA tends to 0 as θ approaches π/2 (with H fixed). By periodicity, cos θ is also0 for θ = π/2 + nπ for any integer n. Therefore the tangent is discontinuous,and undefined, at θ = π/2 + nπ. The period of the tangent function is π,tan(θ + π) = tan θ. (11-6)Additional properties of the trigonometric functions are explored in theexercises.1Please reproduce these graphs with a graphing calculator.2Please reproduce the graph with a graphing calculator.Daniel Stump 311.2 DERIVATIVES OF THE TRIG FUNCTIONSThe derivative of sin θ is, by definition,ddθsin θ = limδ→0sin(θ + δ) − sin θδ. (11-7)eq:defderTo evaluate the limit, simplify the right-hand side by applying the identitysin(A + B) = sin A cos B + cos A sin B; (11-8)letting A = θ and B = δ givessin(θ + δ) = sin θ cos δ + cos θ sin δ. (11-9)Now take the limit δ → 0. The factor cos δ may be approximated by 1, andsin δ may be approximated by δ. (These crucial approximations are explainedbelow.) Making these approximations, (eq:defder11-7) becomesddθsin θ = limδ→0sin θ + cos θ ·δ −sin θδ= limδ→0cos θ = cos θ. (11-10)The result isddθsin θ = cos θ. (11-11)eq:dsinthBefore proceeding, let’s make sure we understand the approximationscos δ ≈ 1 and sin δ ≈ δ, (11-12)which are valid for small δ, i.e., δ 1. Figurefig:sincos?? shows a graph of cos θ andit is obvious from the graph that cos 0 = 1, and cos δ ≈ 1 for small δ. Figurefig:sincos?? also shows sin θ. It is obvious from the graph that sin 0 = 0, and sin δ isapproximately linear in δ for small δ. In fact, the slope of sin θ is 1 at θ = 1,so sin δ ≈ δ for small δ. Figurefig:sinoverx?? shows a graph of sin θ/θ, and it is obviousthat sin θ/θ → 1 as θ → 0; that is, sin δ ≈ δ for small δ. But these analysesare merely numerical. A rigorous mathematical proof that sin δ approachesδ as δ → 0 is given in Appendix X.We have been denoting the independent variable by θ, because θ is a com-mon notation for an angular variable in applications. But since the domain ofthe trig functions is (−∞, +∞) we could just as well use the generic symbolx. Then the derivative formula (eq:dsinth11-11) becomesddxsin x = cos x. (11-13)eq:dsin4 Chapter 11f(x) df/dx f(x) df/dxsin x cos x csc x −cot x csc xcos x −sin x sec x tan x sec xtan x sec2x cot x −csc2xTable 11.1: Derivatives of the trigonometric functions tbl:dtNext, what is the derivative of cos θ? We could go back to the basicdefinition,3but it is easier to use the fact that cos θ is simply related to sin θ,bycos θ = sinπ2− θ. (11-14)eq:compleFor example, in the right triangle in Fig.fig:rightT??, the side adjacent to θ is the sideopposite to the complementary angle ψ ≡ π/2 − θ; socos θ =AHand sin ψ =AH,which implies (eq:comple11-14). [Similarly, sin θ = cos(π/2 −θ).] Now, to differentiatecos θ, apply the chain rule to the identity (eq:comple11-14),ddθcos θ =ddθsinπ2− θ=ddu(sin u) ×dudθwhere u =π2− θ= cos u × (−1) = −cos(π/2 − θ) = −sin θ. (11-15)Hence the derivative of the cosine function isddxcos x = −sin x. (11-16)eq:dcosTabletbl:dt11.1 records the derivatives of the trigonometric functions, startingwith the sine and cosine functions.4The derivatives of the other functionscan be derived by first expressing the function in terms of sine and cosine,and then applying general methods of differentiation, as in the next twoexamples.3See
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