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LECTURE 11: MORE ON TRIGONOMETRIC FUNCTIONS 1. Further standard trigonometric functions Having defined the functions cos : R Ñ r´1, 1s and sin : R Ñ r´1, 1s in the previous lecture, one may easily construct other standard trigonometric functions. Definition. The secant function sec : Rztpk ` 1{2qπ : k P Zu Ñ R and tangent function tan : Rztpk ` 1{2qπ : k P Zu Ñ R are given by sin x 1 , tan x :“ for x P Rztpk ` 1{2qπ : k P Zu. sec x :“ cos x cos x Definition. The cosecant function csc : Rztkπ : k P Zu Ñ R and cotangent function cot : Rztkπ : k P Zu Ñ R are given by cos x 1 , cot x :“ for x P Rztkπ : k P Zu. csc x :“ sin x sin x By using the usual rules for differentiation and the established formulae for cos1 and sin1 , one may easily prove the following theorem. Theorem. For x P Rztpk ` 1{2qπ : k P Zu one has sec1 pxq “ secpxq tanpxq, tan1 pxq “ sec2 pxq. For x P Rztkπ : k P Zu one has csc1 pxq “ ´ cscpxq cotpxq, cot1 pxq “ ´ csc2 pxq. It is, of course, also useful to study inverse trigonometric functions. Clearly sin and cos are not bijective on the whole of R and so one must restrict the domain to ensure the inverses are defined. A standard choice of (maximal) domains is given by sin : r´π{2, π{2s Ñ r´1, 1s, cos : r0, πs Ñ r´1, 1s, tan : p´π{2, π{2q Ñ R. Each of these functions can easily be seen to be bijective (for instance, the derivatives of sin and tan are strictly positive on p´π{2, π{2q). Definition. ‚ The inverse of cos : r0, πs Ñ r´1, 1s is denoted arccos. Hence arccos : r´1, 1s Ñ r0, πs satisfies arccospcos xq “ x for all x P r0, πs. ‚ The inverse of sin : r´π{2, π{2s Ñ r´1, 1s is denoted arcsin. Hence arcsin : r´1, 1s Ñ r´π{2, π{2s satisfies arcsinpsin xq “ x for all x P r´π{2, π{2s. ‚ The inverse of tan : p´π{2, π{2q Ñ r´1, 1s is denoted arctan. Hence arctan : R Ñ p´π{2, π{2q satisfies arctanptan xq “ x for all x P p´π{2, π{2q. 1 2 LECTURE 11: MORE ON TRIGONOMETRIC FUNCTIONS Recall that we defined cos to be the inverse of twice the area function A in the last lecture. Hence, recalling the integral formula for A, it follows that ż1a a arccospxq “ 2Apxq “ x 1 ´ x2 ` 2 1 ´ t2 dt. x 1 We also previously computed the derivative A on p´1, 1q and from our analysis it follows that ´1 arccos1 pxq “ ? for x P p´1, 1q. 1 ´ x2 Since sin1 and tan1 are strictly positive on p´π{2, π{2q, it follows from the theory of inverse functions that arcsin is differentiable on p´1, 1q and arctan is differentiable on R. Theorem. If x P p´1, 1q, then ´1 , 1 ´ x2 1 arcsin1 pxq “ ? . 1 ´ x2 arccos1 pxq “ ? If x P R, then arctan1 pxq “ 1 . 1 ` x2 2. The double angle formula We conclude this discussion of trigonometric functions with a calculus-based proof of the familiar double angle formulae. Theorem. For x, y P R one has cospx ` yq “ cos x cos y ´ sin x sin y, sinpx ` yq “ sin x cos y ` cos x sin y. Our proof is based on elementary theory of differential equations. Lemma. Suppose f : R Ñ R is everywhere twice differentiable and that f 2 ` f “ 0, f p0q “ 0, f 1 p0q “ 0. Then f “ 0. Proof. Multiplying the equation through by 2f 1 it follows that pf 12 ` f 2 q1 “ 2f 2 f 1 ` 2f 1 f “ 0 and therefore f 12 ` f 2 is constant. The conditions f p0q “ f 1 p0q “ 0 imply that f 12 ` f 2 “ 0 and so f “ 0. This lemma is a special case of the following more general theorem. Theorem. Suppose f : R Ñ R is everywhere twice differentiable and that f 2 ` f “ 0, f p0q “ a, f 1 p0q “ b. Then f “ a cos `b sin. LECTURE 11: MORE ON TRIGONOMETRIC FUNCTIONS 3 Jonathan Hickman, Department of mathematics, University of Chicago, 5734 S. University Avenue, Eckhart hall Room 414, Chicago, Illinois, 60637. E-mail address: [email protected]