Worksheet 8.1 Advanced Periodic Logarithmic and Exponential Functions

Maths Quest Maths C Year 12 for Queensland Chapter 8 Advanced Periodic Functions WorkSHEET 8.1 1

WorkSHEET 8.1 Advanced Periodic Logarithmic and Exponential Functions

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1 Using de Moivre’s theorem and the binomial z  cos  i sin expansion, prove that z 2  cos  isin 2 cos2   cos2   sin 2   2cos2  1 Using de Moivre’s theorem, z 2  cos 2  i sin 2 Writing the binomial expansion of z 2 , we have z 2  cos2   sin 2   2i cos sin  cos 2  cos2   sin 2  Applying the Pythagorean Identity, sin 2   1 cos2   cos 2  cos2   1 cos2    2cos2  1  cos 2  cos2   sin 2   2cos2  1

2 Using the multiple angle formulas, prove that 2sin 4x cos 2x

2sinx cos  x = sin  x  sin   x 1 4 4 1 2 2  2 z  z   z  z   2i 2 1  z 4  z 4 z 2  z 2  2i 1  z 6  z 2  z 2  z 6  2i 1  z 6  z 6  z 2  z 2  2i z 6  z 6 z 2  z 2   2i 2i  sin 6x  sin 2x Maths Quest Maths C Year 12 for Queensland Chapter 8 Advanced Periodic Functions WorkSHEET 8.1 2

1 3 4 1 z  z Prove that sin x  cos4x 4cos2x 3 sin x  8 2i 4 z  z 1 sin 4 x    16 4 z  z 1   z 4  4z 3.z 1  6z 2 .z 2  4z.z 3  z 4  z 4  z 4  4z 2  z 2  6  2cos 4x  4 2cos 2x  6  2cos 4x  8cos 2x  6 1 sin 4 x  2cos 4x  8cos 2x  6 16 2  cos 4x  4cos 2x  3 16 1  cos 4x  4cos 2x  3 8

4 2 i 2i Express in standard form. 2e 3 2e 3 2 2  2cos  2isin 3 3  1  3  2    2i   2  2  1 i 3

5 If u  3  i and w  1 i, u  3  i, w  1 i (a) express both u and w in Euler’s form. (a) u  2 u is a complex number in the 4th quadrant of the complex plane   argu   6 i   u  2e 6 w  2 w is in the first quadrant of the complex plane  arg w   4 i   w  2 e 4

u 3 (b) express in standard form. (b) w5 Maths Quest Maths C Year 12 for Queensland Chapter 8 Advanced Periodic Functions WorkSHEET 8.1 3

i u 3 23 e 2  w5 5 5i 2 2 e 4 i 8e 2  3i  4 2 e 4 i 2  e 4 2 2  1 i      2  2 2   1 i

5 (c) find values for m and n such that (c) (Cont.) um w n  8 i . u m wn  8i LHS  u m wn mi n ni  2m e 6 .2 2 e 4 n  n m  m i     2 2.e  4 6  n  3n2m  m i    2 2.e  12  i RHS  23 e 2 n m  23  2 2 n i.e. 3  m  2 2m  n  6 3n  2m 1 and   2k for an integer k. 12 2 Maths Quest Maths C Year 12 for Queensland Chapter 8 Advanced Periodic Functions WorkSHEET 8.1 4

i.e. 3n  2m  6  24k and n  2m  6  4n  12  24k n  3  6k  2m  6  n  6  3  6k 2m  3  6k i.e. m  1.5  3k There is an infinite solution set given by m  1.5  3k and n  3  6k for integer k. Maths Quest Maths C Year 12 for Queensland Chapter 8 Advanced Periodic Functions WorkSHEET 8.1 5

6 Apply Euler’s formula to evaluate  e x sin2x dx x  esin x d x e2ix  cos 2x  isin 2x Ime2ix  sin 2x   e x sin2x dx   e x Ime2ix dx   Ime x .e2ix dx   Ime12ix dx e12ix   Im   1 2i  e12ix 1 2i   Im    1 2i 1 2i  e12ix 1 2i  Im   5  x e 2ix   Im .e 1 2i  5  e x   Im cos 2x  isin 2x1 2i  5  e x   Im cos 2x  2i cos 2x  i sin 2x  2sin 2x  5  e x ie x   Im cos 2x  2sin 2x sin 2x  2cos 2x  5 5     e x sin2x dx e x  sin 2x  2cos 2x c 5

7 (a) Sketch the function y= ex cos( x) over (a) Here, cos x is squeezed between the x the domain  x   . envelopes  e . Graph y  ex and then squeeze y  cos x between the envelopes Maths Quest Maths C Year 12 for Queensland Chapter 8 Advanced Periodic Functions WorkSHEET 8.1 6

x (b) Determine lim e cosx x

(b) limex cosx  0 ex converges rapidly to zero while cos x oscillates between 1. Hence ex cosx oscillates towards y  0 as x increases.

7   (Cont.) (c) Evaluate  ex cosxdx (c)  ex cos x dx 0 0 Consider I   ex cos x dx Apply the parts formula.  d  I  ex  sin x dx   dx  d  ex sin x  ex sin x dx  dx  ex sin x   ex sin x dx Now consider  ex sin x dx d ex sin x dx  ex  cos x dx   dx  ex cos x   ex cos x dx  ex cos x  I i.e. I  ex sin x  ex cos x  I  2I  ex sin x  ex cos x  ex sin x  cos x 1  I  ex sin x  cos x 2 Hence, Maths Quest Maths C Year 12 for Queensland Chapter 8 Advanced Periodic Functions WorkSHEET 8.1 7

 n ex cos x dx  lim ex cos x dx  n  0 0    ex cos x dx 0 1 n  lim  e sin n  cos n n2 1 0   e sin0 cos0 2  1  0   1 2 1  . 2

8 (a) Find the set of complex numbers zn (a) n 3 where n = 1, 2 and 3 given that  2ik  zn  exp  3    n  2ik  k1  3  z  exp  n    n  1, 2, 3. k1  3  1  2i  z1  exp  k1  3  2i  e 3 1 3  z    i 1 2 2 2  2ik 3    z2  exp  k1  3  2i 16i  e 3  e 3 1 3  1 3       i    i 2 2  2 2 

 z2  1 3  2ik 3    z3  exp  k1  3   1 3   1 3      18i    i    i  e  2 2   2 2   11  0  1 3   zn    i, 1, 0  2 2 

(b) Graph zn in the complex plane joining (b) Plot each point in the sequence in the Maths Quest Maths C Year 12 for Queensland Chapter 8 Advanced Periodic Functions WorkSHEET 8.1 8

the points together to form a closed complex plane. Join them together. The figure. What shape is this figure? figure formed is an equilateral triangle.

9 Show that n n sink  sink   Imei  e2i  e3i ⋯⋯ ein  k1 k1  sin  sin2  sin3   sinn  ei  cos  i sin Imei   sin Likewise Imeik   sin k n sin k k1  Imei  Ime2i  Ime3i   Imeni   Imei  e2i  e3i   eni 

10 Given that x  et cosat represents the displacement of a particle at time t, (a) show that x˙t  et cosat asinat (a) x  et cosat dx  x˙ dt dx d  et cosat et cosat dt dt  et cosat aet sinat  x˙  et cosat asinat

(b) If x˙    0 show that (b) 1 x˙   0 tana   , a  0 a  e cosa  asina   0 i.e. cosa  asina   0  asina   cosa  1  tana   a provided a  0 Maths Quest Maths C Year 12 for Queensland Chapter 8 Advanced Periodic Functions WorkSHEET 8.1 9

1 (c) Solve tana   . (c) By making appropriate use of graphics a calculator functions, find the first positive HINT: Use a ‘solver’ routine in the 1 value of a for which tan a   . graphics calculated to show that a a  0.383.