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(i)(1/2)-(-i)(2) 2i Section 12 The Exponential and Trigonometric Functions 69 Notice from both these examples that sines and cosines of complex numbers may be greater than 1. As we shall see (Section 15), although | sin z| < 1 and cos r< 1 for real r, when z is a complex number, sin z and cos z can have any value we like. Using the definitions (11.4) of sin z and cos z, you can show that the familiar trigonometric identities and calculus formulas hold when 0 is replaced by z Example 3. Prove that sin2 z + cos2 z = 1. e2iz-2e-2iz e-eiz2 sin z= 2i e2iz +2+e-2iz cos z sin zcos z = Example 4. Using the definitions (11.4), verify that (d/dz) sinz = cos z. eize ie) ei sin z= PROBLEMS, SECTION 11 Define sin z and cos z by their power series. Write the power series for e. By comparing these series obtain the definition (11.4) of sinz and cos z. Solve the equations e = cos0isin 0, e-i cos 0-isin 0, for cos 0 and sin 0 and so obtain equations (11.3). 1. 2. Find each of the following in rectangular form r+iy and check your results by computer. Remember to save time by doing as much as you can in your head (in/4)+(Ia 2)/2 e-(in/4)+ln 3 3 la 2-i 4 3 6. cos(in 5) 9. sin(-iIn 3) cos(-2i In 3) 7. tan(in 2) 10. sin(iIn i In the following integrals express the sines and cosines in exponential form and then integrate to show that: Cos2 3r drT cOs 2z cos 3r dr=0 sin 4r dr=T 13. sin 2r sin 3r dr= 0 14. 15. sin 2r cos 3 dz = 0 sin 3r cos 4r dr= 0 Evaluate fea+ib)adz and take real and imaginary parts to show that: ,ar Cos br dr= a cos br + b sin br) a2b2 a(a sin br- b cos br) a2 +b2

(i)(1/2)-(-i)(2) 2i Section 12 The Exponential and Trigonometric Functions 69 Notice from both these examples that sines and cosines of complex numbers may be greater than 1. As we shall see (Section 15), although | sin z| < 1 and cos r< 1 for real r, when z is a complex number, sin z and cos z can have any value we like. Using the definitions (11.4) of sin z and cos z, you can show that the familiar trigonometric identities and calculus formulas hold when 0 is replaced by z Example 3. Prove that sin2 z + cos2 z = 1. e2iz-2e-2iz e-eiz2 sin z= 2i e2iz +2+e-2iz cos z sin zcos z = Example 4. Using the definitions (11.4), verify that (d/dz) sinz = cos z. eize ie) ei sin z= PROBLEMS, SECTION 11 Define sin z and cos z by their power series. Write the power series for e. By comparing these series obtain the definition (11.4) of sinz and cos z. Solve the equations e = cos0isin 0, e-i cos 0-isin 0, for cos 0 and sin 0 and so obtain equations (11.3). 1. 2. Find each of the following in rectangular form r+iy and check your results by computer. Remember to save time by doing as much as you can in your head (in/4)+(Ia 2)/2 e-(in/4)+ln 3 3 la 2-i 4 3 6. cos(in 5) 9. sin(-iIn 3) cos(-2i In 3) 7. tan(in 2) 10. sin(iIn i In the following integrals express the sines and cosines in exponential form and then integrate to show that: Cos2 3r drT cOs 2z cos 3r dr=0 sin 4r dr=T 13. sin 2r sin 3r dr= 0 14. 15. sin 2r cos 3 dz = 0 sin 3r cos 4r dr= 0 Evaluate fea+ib)adz and take real and imaginary parts to show that: ,ar Cos br dr= a cos br + b sin br) a2b2 a(a sin br- b cos br) a2 +b2

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter8: Complex Numbers And Polarcoordinates
Section: Chapter Questions
Problem 1RP: As was mentioned in the introduction to this chapter, Jerome Cardan (Girolamo Cardano) was unable to...
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(i)(1/2)-(-i)(2) 2i Section 12 The Exponential and Trigonometric Functions 69 Notice from both these examples that sines and cosines of complex numbers may be greater than 1. As we shall see (Section 15), although | sin z| < 1 and cos r< 1 for real r, when z is a complex number, sin z and cos z can have any value we like. Using the definitions (11.4) of sin z and cos z, you can show that the familiar trigonometric identities and calculus formulas hold when 0 is replaced by z Example 3. Prove that sin2 z + cos2 z = 1. e2iz-2e-2iz e-eiz2 sin z= 2i e2iz +2+e-2iz cos z sin zcos z = Example 4. Using the definitions (11.4), verify that (d/dz) sinz = cos z. eize ie) ei sin z= PROBLEMS, SECTION 11 Define sin z and cos z by their power series. Write the power series for e. By comparing these series obtain the definition (11.4) of sinz and cos z. Solve the equations e = cos0isin 0, e-i cos 0-isin 0, for cos 0 and sin 0 and so obtain equations (11.3). 1. 2. Find each of the following in rectangular form r+iy and check your results by computer. Remember to save time by doing as much as you can in your head (in/4)+(Ia 2)/2 e-(in/4)+ln 3 3 la 2-i 4 3 6. cos(in 5) 9. sin(-iIn 3) cos(-2i In 3) 7. tan(in 2) 10. sin(iIn i In the following integrals express the sines and cosines in exponential form and then integrate to show that: Cos2 3r drT cOs 2z cos 3r dr=0 sin 4r dr=T 13. sin 2r sin 3r dr= 0 14. 15. sin 2r cos 3 dz = 0 sin 3r cos 4r dr= 0 Evaluate fea+ib)adz and take real and imaginary parts to show that: ,ar Cos br dr= a cos br + b sin br) a2b2 a(a sin br- b cos br) a2 +b2
Transcribed Image Text:(i)(1/2)-(-i)(2) 2i Section 12 The Exponential and Trigonometric Functions 69 Notice from both these examples that sines and cosines of complex numbers may be greater than 1. As we shall see (Section 15), although | sin z| < 1 and cos r< 1 for real r, when z is a complex number, sin z and cos z can have any value we like. Using the definitions (11.4) of sin z and cos z, you can show that the familiar trigonometric identities and calculus formulas hold when 0 is replaced by z Example 3. Prove that sin2 z + cos2 z = 1. e2iz-2e-2iz e-eiz2 sin z= 2i e2iz +2+e-2iz cos z sin zcos z = Example 4. Using the definitions (11.4), verify that (d/dz) sinz = cos z. eize ie) ei sin z= PROBLEMS, SECTION 11 Define sin z and cos z by their power series. Write the power series for e. By comparing these series obtain the definition (11.4) of sinz and cos z. Solve the equations e = cos0isin 0, e-i cos 0-isin 0, for cos 0 and sin 0 and so obtain equations (11.3). 1. 2. Find each of the following in rectangular form r+iy and check your results by computer. Remember to save time by doing as much as you can in your head (in/4)+(Ia 2)/2 e-(in/4)+ln 3 3 la 2-i 4 3 6. cos(in 5) 9. sin(-iIn 3) cos(-2i In 3) 7. tan(in 2) 10. sin(iIn i In the following integrals express the sines and cosines in exponential form and then integrate to show that: Cos2 3r drT cOs 2z cos 3r dr=0 sin 4r dr=T 13. sin 2r sin 3r dr= 0 14. 15. sin 2r cos 3 dz = 0 sin 3r cos 4r dr= 0 Evaluate fea+ib)adz and take real and imaginary parts to show that: ,ar Cos br dr= a cos br + b sin br) a2b2 a(a sin br- b cos br) a2 +b2
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