Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Trigonometric and Hyperbolic Functions Bernd Schroder¨ logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions 1. For real numbers q we have eiq = cos(q) + isin(q). 2. Replacing q with −q we obtain e−iq = cos(q) − isin(q). (Remember that the cosine is even and the sine is odd.) eiq + e−iq 3. Adding the two and dividing by 2 gives cos(q) = . 2 eiq − e−iq 4. Subtracting the two and dividing by 2i gives sin(q) = . 2i 5. The right sides above make sense for all complex numbers. Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Introduction logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions 2. Replacing q with −q we obtain e−iq = cos(q) − isin(q). (Remember that the cosine is even and the sine is odd.) eiq + e−iq 3. Adding the two and dividing by 2 gives cos(q) = . 2 eiq − e−iq 4. Subtracting the two and dividing by 2i gives sin(q) = . 2i 5. The right sides above make sense for all complex numbers. Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Introduction 1. For real numbers q we have eiq = cos(q) + isin(q). logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions (Remember that the cosine is even and the sine is odd.) eiq + e−iq 3. Adding the two and dividing by 2 gives cos(q) = . 2 eiq − e−iq 4. Subtracting the two and dividing by 2i gives sin(q) = . 2i 5. The right sides above make sense for all complex numbers. Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Introduction 1. For real numbers q we have eiq = cos(q) + isin(q). 2. Replacing q with −q we obtain e−iq = cos(q) − isin(q). logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions eiq + e−iq 3. Adding the two and dividing by 2 gives cos(q) = . 2 eiq − e−iq 4. Subtracting the two and dividing by 2i gives sin(q) = . 2i 5. The right sides above make sense for all complex numbers. Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Introduction 1. For real numbers q we have eiq = cos(q) + isin(q). 2. Replacing q with −q we obtain e−iq = cos(q) − isin(q). (Remember that the cosine is even and the sine is odd.) logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions eiq − e−iq 4. Subtracting the two and dividing by 2i gives sin(q) = . 2i 5. The right sides above make sense for all complex numbers. Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Introduction 1. For real numbers q we have eiq = cos(q) + isin(q). 2. Replacing q with −q we obtain e−iq = cos(q) − isin(q). (Remember that the cosine is even and the sine is odd.) eiq + e−iq 3. Adding the two and dividing by 2 gives cos(q) = . 2 logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions 5. The right sides above make sense for all complex numbers. Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Introduction 1. For real numbers q we have eiq = cos(q) + isin(q). 2. Replacing q with −q we obtain e−iq = cos(q) − isin(q). (Remember that the cosine is even and the sine is odd.) eiq + e−iq 3. Adding the two and dividing by 2 gives cos(q) = . 2 eiq − e−iq 4. Subtracting the two and dividing by 2i gives sin(q) = . 2i logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Introduction 1. For real numbers q we have eiq = cos(q) + isin(q). 2. Replacing q with −q we obtain e−iq = cos(q) − isin(q). (Remember that the cosine is even and the sine is odd.) eiq + e−iq 3. Adding the two and dividing by 2 gives cos(q) = . 2 eiq − e−iq 4. Subtracting the two and dividing by 2i gives sin(q) = . 2i 5. The right sides above make sense for all complex numbers. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions For any complex number z we define eiz + e−iz cos(z) = 2 and eiz − e−iz sin(z) = : 2i Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Definition. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions and eiz − e−iz sin(z) = : 2i Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Definition. For any complex number z we define eiz + e−iz cos(z) = 2 logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Definition. For any complex number z we define eiz + e−iz cos(z) = 2 and eiz − e−iz sin(z) = : 2i logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions 1. sin(z1 + z2) = sin(z1)cos(z2) + cos(z1)sin(z2) 2. cos(z1 + z2) = cos(z1)cos(z2) − sin(z1)sin(z2) 3. sin2(z) + cos2(z) = 1 4. sin(z + 2p) = sin(z) 5. cos(z + 2p) = cos(z) Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions This Time, All Common Identities Carry Over logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions 2. cos(z1 + z2) = cos(z1)cos(z2) − sin(z1)sin(z2) 3. sin2(z) + cos2(z) = 1 4. sin(z + 2p) = sin(z) 5. cos(z + 2p) = cos(z) Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions This Time, All Common Identities Carry Over 1. sin(z1 + z2) = sin(z1)cos(z2) + cos(z1)sin(z2) logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions 3. sin2(z) + cos2(z) = 1 4. sin(z + 2p) = sin(z) 5. cos(z + 2p) = cos(z) Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions This Time, All Common Identities Carry Over 1. sin(z1 + z2) = sin(z1)cos(z2) + cos(z1)sin(z2) 2. cos(z1 + z2) = cos(z1)cos(z2) − sin(z1)sin(z2) logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions 4. sin(z + 2p) = sin(z) 5. cos(z + 2p) = cos(z) Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions This Time, All Common Identities Carry Over 1. sin(z1 + z2) = sin(z1)cos(z2) + cos(z1)sin(z2) 2. cos(z1 + z2) = cos(z1)cos(z2) − sin(z1)sin(z2) 3. sin2(z) + cos2(z) = 1 logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions 5. cos(z + 2p) = cos(z) Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions This Time, All Common Identities Carry Over 1. sin(z1 + z2) = sin(z1)cos(z2) + cos(z1)sin(z2) 2. cos(z1 + z2) = cos(z1)cos(z2) − sin(z1)sin(z2) 3. sin2(z) + cos2(z) = 1 4. sin(z + 2p) = sin(z) logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions This Time, All Common Identities Carry Over 1. sin(z1 + z2) = sin(z1)cos(z2) + cos(z1)sin(z2) 2. cos(z1 + z2) = cos(z1)cos(z2) − sin(z1)sin(z2) 3. sin2(z) + cos2(z) = 1 4. sin(z + 2p) = sin(z) 5. cos(z + 2p) = cos(z) logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions sin(z1)cos(z2) + cos(z1)sin(z2) eiz1 − e−iz1 eiz2 + e−iz2 eiz1 + e−iz1 eiz2 − e−iz2 = + 2i 2 2 2i 1 = ei(z1+z2) + ei(z1−z2) − ei(−z1+z2) − e−i(z1+z2) 4i + ei(z1+z2) − ei(z1−z2) + ei(−z1+z2) − e−i(z1+z2) 1 ei(z1+z2) − e−i(z1+z2) = 2ei(z1+z2) − 2e−i(z1+z2) = 4i 2i = sin(z1 + z2) This is why many people like to work with the complex definition of the trigonometric functions. Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Some Proofs Actually Are Simpler Now logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions eiz1 − e−iz1 eiz2 + e−iz2 eiz1 + e−iz1 eiz2 − e−iz2 = + 2i 2 2 2i 1 = ei(z1+z2) + ei(z1−z2) − ei(−z1+z2) − e−i(z1+z2) 4i + ei(z1+z2) − ei(z1−z2) + ei(−z1+z2) − e−i(z1+z2) 1 ei(z1+z2) − e−i(z1+z2) = 2ei(z1+z2) − 2e−i(z1+z2) = 4i 2i = sin(z1 + z2) This is why many people like to work with the complex definition of the trigonometric functions. Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Some Proofs Actually Are Simpler Now sin(z1)cos(z2) + cos(z1)sin(z2) logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Trigonometric and Hyperbolic Functions 1 = ei(z1+z2) + ei(z1−z2) − ei(−z1+z2) − e−i(z1+z2) 4i + ei(z1+z2) − ei(z1−z2) + ei(−z1+z2) − e−i(z1+z2) 1 ei(z1+z2) − e−i(z1+z2) = 2ei(z1+z2) − 2e−i(z1+z2) = 4i 2i = sin(z1 + z2) This is why many people like to work with the complex definition of the trigonometric functions.