Lecture Notes Di erentiating Trigonometric Functions page 1

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1 Lecture Notes Di erentiating Trigonometric Functions age (sin ) 7 sin () sin 8 cos 3 (tan ) sec tan + 9 tan + 4 (cot ) csc cot 0 cot + 5 sin (sec ) cos sec tan sec jj 6 (csc ) sin csc cot csc jj c Hiegkuti, Powell, 0 Last revise: January 9, 07

2 Lecture Notes Di erentiating Trigonometric Functions age Proofs Teorems an : (sin ) an () sin sin Claim ) lim!0 Proof: Tis teorem an te net one are necessary for i erentiating sin an Recall a teorem: Let r be te raius of a circle If is measure in raians, ten te area of a sector wit a central angle of is A sector r (Notation: AB will enote te lengt of line segment AB) Let be a very small ositive angle, measure in raians, rawn into a unit circle as sown on te icture below Let B be te oint were te unit circle intersects te ray etermine by We ten raw a tangent line to te circle at oint B Let A be te oint were te tangent line intersects te ais We also raw a vertical line troug B: Let D be te oint were tis vertical line intersects te ais Finally, let us enote by E te oint wit coorinates (0; ) Te roof will be base on te following fact: because tey inclue eac oter, te following tree areas can be easily comare: Area of triangle CDB Area of sector CEB Area of triangle ABC Area of triangle CDB: te orizontal sie, CD an te vertical sie, DB sin Since tis is a rigt triangle, te area is: A CDB sin Area of sector CEB: A sector Area of triangle ABC: tere is a rigt angle at oint B because te tangent line rawn to a circle is erenicular to te raius rawn to te oint of tangency So te area is A ABC AB BC Clearly BC To comute AB, in triangle ABC, tan AB an so AB tan tan sin Area of triangle ABC: () (tan ) or So now Area of triangle CDB Area of sector CEB Area of triangle ABC translates to sin sin c Hiegkuti, Powell, 0 Last revise: January 9, 07

3 Lecture Notes Di erentiating Trigonometric Functions age 3 Let us ivie all tree sies by sin reverse te inequality signs Suose now tat aroaces zero Because is small an ositive, sin te quantity locke in between tose two must also aroac : is ositive an so we o not nee to sin Ten bot an aroac By te sanwic rincile, sin, sin # # If sin aroaces ; so is its recirocal, sin sin So far, we ave roven te statement for ositive values of, tat is, lim!0 + for negative values of A similar argument works Claim ) lim 0!0 Proof: lim!0 lim lim + cos lim!0!0 +!0 ( + ) lim!0 sin lim!0 ( + ) lim sin!0 sin + lim sin!0 lim sin! cos ( + ) We are now reay to rove tat Proof: (sin ) lim!0 lim!0 (sin ) an sin ( + ) sin cos () sin sin sin cos + sin sin lim!0 sin sin sin (cos ) + lim + lim sin cos sin sin lim + lim sin 0 + () lim!0 cos ( + ) cos lim cos lim cos sin sin lim!0 sin sin (cos ) lim sin sin lim 0 sin sin sin sin lim c Hiegkuti, Powell, 0 Last revise: January 9, 07

4 Lecture Notes Di erentiating Trigonometric Functions age 4 Teorem 3 an 4: (tan ) sec tan + an (cot ) csc cot Proof: We write tan sin an aly te quotient rule sin sin sin cos ( sin ) sin cos cos + sin cos cos sec We will now rove cos tan +; wic is a very imortant connection Looking at te revious comutation, cos cos + sin cos cos cos + sin cos + tan Te roof for (cot ) cot csc is very similar We aly te quotient rule sin sin sin sin () (cot ) sin sin sin sin csc Also, sin cos sin sin sin cos sin cot sin cos sin Teorems 5 an 6: (sec ) sec tan an (csc ) csc cot Proof: We write sec () an aly te cain rule (sec ) () () () cos ( sin sin ) cos sin sec tan Te roof for csc is virtually ientical: (csc ) (sin ) we aly te cain rule (sin ) sin sin sin sin sin cot csc Note: wy o we refer te form sec tan over te form sin cos? in i erentiating te inverse functions sec an csc One of te reasons is te aventage we ll see c Hiegkuti, Powell, 0 Last revise: January 9, 07

5 Lecture Notes Di erentiating Trigonometric Functions age 5 Teorems 7 an 8: sin an cos Proof: Recall tat wen we comose a function f wit its inverse f ; te result is always te same function, (also calle te ientity function, i () ) f f () We will state tis fact for f () sin an i erentiate bot sies of te equation For te left-an sie, we use te cain rule cos sin We now nee to simlify cos sin sin sin sin ivie by cos sin sin cos sin We will resent two metos to simlify tis eression Meto We rst introuce a new variable, Let sin Tis means tat We nee to simlify cos sin cos Since sin + cos ; {z } q cos sin Since < <, cos is ositive an so cos sin Meto We rst introuce a new variable, Let sin Tis means tat Now te goal is to simlify cos sin cos {z } We will use a rigt triangle to n te eression - u to its sign We rst raw a rigt triangle in wic sin < <, an sin < <, an sin Te aventage of tis meto is tat now we can rea any trigonometric function value of sin using tis rigt triangle Net we use te Pytagoream Teorem to n te missing sie to be From te triangle, cos cos sin Te answer at tis oint is really as te triangle gave us te answer only u to a sign For te sign, we nee to argue using te location of on te unit circle Since sin, Tus is in te rst or in te fourt quarant In bot cases, cosine is ositive, tus cos sin Tus sin cos sin c Hiegkuti, Powell, 0 Last revise: January 9, 07

6 Lecture Notes Di erentiating Trigonometric Functions age 6 Te roof for Proof: cos is virtually ientical Recall tat wen we comose a function f wit its inverse f ; te result is always te same function f f () We will state tis fact for f () an i erentiate bot sies of te equation use te cain rule For te left-an sie, we sin cos cos cos cos ivie by sin cos cos We now nee to simlify te eression sin cos sin (cos ) Meto Let cos Ten cos an is between 0 an! sin cos {z } We will resent two metos for tis sin cos Since is between 0 an, sin is ositive an so sin Meto Let cos Ten cos an is between 0 an We rst raw a triangle in wic cos Please note tat every time we aroac suc a trigonometric question using a rigt triangle, our answer woul be accurate u to sign - for te sign we woul ave to argue searately Now we can rea any trigonometric function value using tis triangle Now we rea sine from te triangle: We n te missing sie via te Pytagoream Teorem: sin sin cos Te answer at tis oint is really as te triangle gave us te answer only u to a sign For te sign, we nee to argue using te location of on te unit circle Since cos, 0 Tus is in te rst or in te secon quarant In bot cases, sine is ositive, tus sin cos Consequently, If cos Enricment sin (cos ) sin an cos are oosites, ten wat can be sai about te function f () sin +cos? c Hiegkuti, Powell, 0 Last revise: January 9, 07

7 Lecture Notes Di erentiating Trigonometric Functions age 7 Teorems 9 an 0: tan + an cot + Proof: Recall tat (tan ) sec tan + inverse f ; te result is always te same function Also recall tat wen we comose a function f wit its f f () We will state tis fact for f () tan an i erentiate bot sies of te equation use te cain rule For te left-an sie, we tan tan tan sec tan tan tan + tan tan tan + tan ivie by + tan + Te roof for cot cot an i erentiate + is virtually ientical Recall tat cot cot We comose te function cot wit its inverse cot cot cot cot cot cot cot cot ivie by cot + Teorem an : sec jj an csc jj Proof: We comose te function sec wit its inverse sec an i erentiate Recall tat (sec ) sec tan sec sec sec sec tan sec sec sec sec tan sec sec ivie by tan sec sec tan (sec ) We now just nee to simlify te eression tan sec c Hiegkuti, Powell, 0 Last revise: January 9, 07

8 Lecture Notes Di erentiating Trigonometric Functions age 8 Meto To simlify tan sec, we introuce a new variable Let sec Ten we ave tan sec{z } tan were sec an is between 0 an Recall tat sec tan + If we on t ave tis formula memorize, we can easily erive it from te Pytagorean ientity sin + cos ivie by cos sin cos + cos cos cos tan + sec tan sec Meto To simlify tan sec, we introuce a new variable Let sec Ten sec an is between 0 an : Ten we nee to comute tan We raw a rigt triangle in wic sec Now we can rea any trigonometric function value using tis triangle We n te missing sie via te Pytagoream Teorem: Now we rea from te triangle: tan tan sec Te answer at tis oint is really as te triangle gave us te answer only u to a sign Tus te erivative is sec We now nee to gure out te sign of te erivative From te gra of sec we can see tat it is strictly increasing on bot intervals making u its omain, tus te erivative is always ositive If is ositive, ten sec an if is negative, ten sec Tis can be eresse in a sorter form as sec jj Te roof for csc its inverse an i erentiate jj Recall tat (csc ) is virtually ientical As before, we comose te function csc wit csc cot csc csc csc csc cot csc csc csc csc cot csc csc ivie by cot csc csc cot (csc ) c Hiegkuti, Powell, 0 Last revise: January 9, 07

9 Lecture Notes Di erentiating Trigonometric Functions age 9 We nee to simlify cot csc Let csc an cot csc {z } cot were csc an is between Meto We start wit te Pytagorean ientity an ivie bot sies by sin sin + cos ivie by sin sin sin + cos sin sin + cot csc cot csc cot csc Meto We raw a rigt triangle in wic csc We n te missing sie using te Pytagorean Teorem an rea te esire trigonometric function value Tus te erivative is csc From te gra of csc we can see tat it is strictly ecreasing on bot intervals of its omain, tus te erivative is always negative If is ositive, ten csc an if is negative, ten csc Tis can be eresse in a sorter form as csc jj For more ocuments like tis, visit our age at tt://wwwteacingmartaiegkuticom an click on Lecture Notes questions or comments to miegkuti@ccceu c Hiegkuti, Powell, 0 Last revise: January 9, 07