1 Finding Trigonometric Derivatives

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1 MTH 121 Fall 2008 Essex County College Division of Matematics Hanout Version 8 1 October 2, Fining Trigonometric Derivatives 1.1 Te Derivative as a Function Te efinition of te erivative as a function is, f f (x + ) f (x) (x) =, 0 provie te it exists. 2 Okay, I m being repetitious ere, but it is noneteless necessary to be remine of tis efinition before proceeing forwar. 1.2 Te Derivative of Sine I like to sow tat te erivative of te sine function is te cosine function. Sowing tat tis is te case is in fact ifficult. But as I always say, tese tings ave alreay been one an are te result of someone else s ar work. Seeing wat oters ave one is important, an will opefully motivate you to o te same. Te explanation an proof outline ere, altoug not etaile, soul be sufficient to get you tinking. However, please take a look at te book s proof too. Work: First, raw one cycle of te sine function, an it s erivative 3 on te same grap Figure 1: Te sine function (soli) an its erivative (ase). 1 Tis ocument was prepare by Ron Bannon using L A TEX 2ε. 2 Now of course our new function will ave a omain tat may iffer from te originating function. Yes, te erivative may not be efine at all points along te function. 3 Same meto tat we re oing in class. Tat is, try to fin te slopes at some points an ten connect te ots. 1

2 You soul observe tat te ase curve looks like te cosine curve, but tis is certainly not a proof, but at least it s giving us a int. To fin te erivative of te sine function we will nee to use te efinition of erivative. sin (x + ) sin x (sin x) = x 0 Tis it oes not look easy, an you may woner wy its being rewritten in tis form. 4 sin (x + ) sin x (sin x) = x 0 sin x cos + sin cos x sin x = 0 ] sin x cos sin x sin cos x = + 0 = sin x cos 1 + cos x sin ] 0 Still not muc progress, but at least I see a simple it witin te bigger problem tat we ve seen before. sin 0 If you can recall, you were aske 5 to o tis problem numerically an foun tat it equale one. Now we nee to sow tis. Here s one way to o so. Tis is a geometric argument, but iffers from te book s geometric approac. 0 Figure 2: Will be iscusse an labele in class. 4 Te main reason is tat tis is wat you i in MTH-120, an sum ientities were extensively use. 5 If you re reaing te book an oing te suggeste review problems, you ve seen tis it! 2

3 We will label tis iagram in class using a, b, c an θ (0, π/2), an iscuss wy a < b. We will get te following relationsip as well, simply by using trigonometry. sin θ < θ sin θ θ < 1 Now fining c, wic simple turns out to be tan θ an using an existing teorem tat states tat θ < tan θ for 0 < θ < π/2. 6 So now we finally ave (tis will also be iscusse in class). sin θ < θ < tan θ for 0 < θ < π/2 θ 1 < < 1 sin θ cos θ 1 > sin θ > cos θ θ Now, using te Squeeze Teorem, we will fin 7 tat. sin θ θ 0 θ = 1 Tat s a lot of work. But as you can see, we also nee. cos θ 1 θ 0 θ Wic will easily be sown, in class, to be equal to zero. Finally, we ave wat we expecte from te beginning. sin (x + ) sin x (sin x) = x 0 sin x cos + sin cos x sin x = 0 ] sin x cos sin x sin cos x = + 0 = sin x cos 1 + cos x sin ] 0 = sin x cos 1 + cos x sin 0 0 cos 1 sin = sin x + cos x 0 0 = sin x 0 + cos x 1 = cos x 6 Tis proof is not easy, see page A43 of your textbook if intereste. 7 We really just sowe tat tis is true from te rigt of zero, but fortunately tis is an even function, so we ve really sowe it from bot te rigt an left. 3

4 1.3 Insigtful Graps Some graps, altoug not proofs, over insigt. 0 Figure 3: Tis is y = tan x an y = x. Certainly, wen one looks at tis grap it may not be so clear tat tan x > x for 0 < x < π/2. Tis is wy we nee to prove tings. 0 Figure 4: Tis is y = sin x x. Now, I really tink tis grap is convincing, but we still nee to prove it as we i toay. 4

5 2 Te Remaining Rules an Examples Sow all work an box te final answer. 1. You soul be able to prove using te efinition of te erivative tat (cos x) = sin x, x 2. Using rules you soul be able to sow tat x (tan x) = sec2 x. 3. Using rules you soul be able to sow tat (csc x) = cot x csc x. x 5

6 4. Using rules you soul be able to sow tat (sec x) = tan x sec x. x 5. Using rules you soul be able to sow tat x (cot x) = csc2 x. 6. Fin an equation of te tangent line to te curve y = sec x 2 cos x at te point (π/3, 1). 7. Differentiate. y = x + sin x x 2 + cos x 6

7 8. Fin te it. sin 2 3x x 0 x 2 9. Fin te it. x 0 1 tan x sin x cos x 10. Suppose f (π/3) = 4 an f (π/3) = 2, an let g (x) = f (x) sin x an (x) = cos x f (x). Fin g (π/3) an (π/3). 11. If f (β) = β sin β, fin f (β) an f (β). 7