Trigonometric and Hyperbolic Functions
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Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Trigonometric and Hyperbolic Functions Bernd Schröder Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Introduction Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Introduction 1. For real numbers θ we have eiθ = cos(θ ) + i sin(θ ). Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Introduction 1. For real numbers θ we have eiθ = cos(θ ) + i sin(θ ). 2. Replacing θ with −θ we obtain e−iθ = cos(θ ) − i sin(θ ). Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Introduction 1. For real numbers θ we have eiθ = cos(θ ) + i sin(θ ). 2. Replacing θ with −θ we obtain e−iθ = cos(θ ) − i sin(θ ). (Remember that the cosine is even and the sine is odd.) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Introduction 1. For real numbers θ we have eiθ = cos(θ ) + i sin(θ ). 2. Replacing θ with −θ we obtain e−iθ = cos(θ ) − i sin(θ ). (Remember that the cosine is even and the sine is odd.) eiθ + e−iθ 3. Adding the two and dividing by 2 gives cos(θ ) = . 2 Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Introduction 1. For real numbers θ we have eiθ = cos(θ ) + i sin(θ ). 2. Replacing θ with −θ we obtain e−iθ = cos(θ ) − i sin(θ ). (Remember that the cosine is even and the sine is odd.) eiθ + e−iθ 3. Adding the two and dividing by 2 gives cos(θ ) = . 2 eiθ − e−iθ . 4. Subtracting the two and dividing by 2i gives sin(θ ) = 2i Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Introduction 1. For real numbers θ we have eiθ = cos(θ ) + i sin(θ ). 2. Replacing θ with −θ we obtain e−iθ = cos(θ ) − i sin(θ ). (Remember that the cosine is even and the sine is odd.) eiθ + e−iθ 3. Adding the two and dividing by 2 gives cos(θ ) = . 2 eiθ − e−iθ . 4. Subtracting the two and dividing by 2i gives sin(θ ) = 2i 5. The right sides above make sense for all complex numbers. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Definition. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Definition. For any complex number z we define cos(z) = Bernd Schröder Trigonometric and Hyperbolic Functions eiz + e−iz 2 logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Definition. For any complex number z we define cos(z) = eiz + e−iz 2 sin(z) = eiz − e−iz . 2i and Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions This Time, All Common Identities Carry Over Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science 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 ) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science 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 ) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science 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 Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions This Time, All Common Identities Carry Over 1. 2. 3. 4. sin(z1 + z2 ) = sin(z1 ) cos(z2 ) + cos(z1 ) sin(z2 ) cos(z1 + z2 ) = cos(z1 ) cos(z2 ) − sin(z1 ) sin(z2 ) sin2 (z) + cos2 (z) = 1 sin(z + 2π) = sin(z) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions This Time, All Common Identities Carry Over 1. 2. 3. 4. 5. sin(z1 + z2 ) = sin(z1 ) cos(z2 ) + cos(z1 ) sin(z2 ) cos(z1 + z2 ) = cos(z1 ) cos(z2 ) − sin(z1 ) sin(z2 ) sin2 (z) + cos2 (z) = 1 sin(z + 2π) = sin(z) cos(z + 2π) = cos(z) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Some Proofs Actually Are Simpler Now Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Some Proofs Actually Are Simpler Now sin(z1 ) cos(z2 ) + cos(z1 ) sin(z2 ) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Some Proofs Actually Are Simpler Now sin(z1 ) cos(z2 ) + cos(z1 ) sin(z2 ) eiz1 − e−iz1 eiz2 + e−iz2 eiz1 + e−iz1 eiz2 − e−iz2 = + 2i 2 2 2i Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Some Proofs Actually Are Simpler Now 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 i(z1 +z2 ) + ei(z1 −z2 ) − ei(−z1 +z2 ) − e−i(z1 +z2 ) e = 4i Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Some Proofs Actually Are Simpler Now 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 i(z1 +z2 ) + ei(z1 −z2 ) − ei(−z1 +z2 ) − e−i(z1 +z2 ) e = 4i + ei(z1 +z2 ) − ei(z1 −z2 ) + ei(−z1 +z2 ) − e−i(z1 +z2 ) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Some Proofs Actually Are Simpler Now 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 i(z1 +z2 ) + ei(z1 −z2 ) − ei(−z1 +z2 ) − e−i(z1 +z2 ) e = 4i + ei(z1 +z2 ) − ei(z1 −z2 ) + ei(−z1 +z2 ) − e−i(z1 +z2 ) 1 i(z1 +z2 ) = 2e − 2e−i(z1 +z2 ) 4i Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Some Proofs Actually Are Simpler Now 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 i(z1 +z2 ) + ei(z1 −z2 ) − ei(−z1 +z2 ) − e−i(z1 +z2 ) e = 4i + ei(z1 +z2 ) − ei(z1 −z2 ) + ei(−z1 +z2 ) − e−i(z1 +z2 ) ei(z1 +z2 ) − e−i(z1 +z2 ) 1 i(z1 +z2 ) = 2e − 2e−i(z1 +z2 ) = 4i 2i Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Some Proofs Actually Are Simpler Now 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 i(z1 +z2 ) + ei(z1 −z2 ) − ei(−z1 +z2 ) − e−i(z1 +z2 ) e = 4i + ei(z1 +z2 ) − ei(z1 −z2 ) + ei(−z1 +z2 ) − e−i(z1 +z2 ) ei(z1 +z2 ) − e−i(z1 +z2 ) 1 i(z1 +z2 ) = 2e − 2e−i(z1 +z2 ) = 4i 2i = sin(z1 + z2 ) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Some Proofs Actually Are Simpler Now 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 i(z1 +z2 ) + ei(z1 −z2 ) − ei(−z1 +z2 ) − e−i(z1 +z2 ) e = 4i + ei(z1 +z2 ) − ei(z1 −z2 ) + ei(−z1 +z2 ) − e−i(z1 +z2 ) ei(z1 +z2 ) − e−i(z1 +z2 ) 1 i(z1 +z2 ) = 2e − 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. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Other Proofs Stay The Same Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Other Proofs Stay The Same sin(z + 2π) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Other Proofs Stay The Same sin(z + 2π) = sin(z) cos(2π) + cos(z) sin(2π) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Other Proofs Stay The Same sin(z + 2π) = sin(z) cos(2π) + cos(z) sin(2π) = sin(z) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Other Trigonometric Functions Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Other Trigonometric Functions 1. tan(z) := sin(z) cos(z) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Other Trigonometric Functions sin(z) cos(z) cos(z) 2. cot(z) := sin(z) 1. tan(z) := Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Other Trigonometric Functions sin(z) cos(z) cos(z) 2. cot(z) := sin(z) 1 3. sec(z) := cos(z) 1. tan(z) := Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Other Trigonometric Functions sin(z) cos(z) cos(z) 2. cot(z) := sin(z) 1 3. sec(z) := cos(z) 1 4. csc(z) := sin(z) 1. tan(z) := Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Other Trigonometric Functions sin(z) cos(z) cos(z) 2. cot(z) := sin(z) 1 3. sec(z) := cos(z) 1 4. csc(z) := sin(z) Note that secant and cosecant are not very common (around the world). 1. tan(z) := Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Derivatives Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Derivatives 1. d sin(z) dz Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Derivatives 1. d d eiz −e−iz sin(z) = dz dz 2i Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Derivatives 1. d d eiz −e−iz ieiz +ie−iz sin(z) = = dz dz 2i 2i Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Derivatives 1. d d eiz −e−iz ieiz +ie−iz eiz +e−iz sin(z) = = = dz dz 2i 2i 2 Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Derivatives 1. d d eiz −e−iz ieiz +ie−iz eiz +e−iz sin(z) = = = = cos(z) dz dz 2i 2i 2 Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Derivatives d d eiz −e−iz ieiz +ie−iz eiz +e−iz sin(z) = = = = cos(z) dz dz 2i 2i 2 d cos(z) = − sin(z) 2. dz 1. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Derivatives d d eiz −e−iz ieiz +ie−iz eiz +e−iz sin(z) = = = = cos(z) dz dz 2i 2i 2 d cos(z) = − sin(z) 2. dz 1 d tan(z) = 3. dz cos2 (z) 1. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Derivatives d d eiz −e−iz ieiz +ie−iz eiz +e−iz sin(z) = = = = cos(z) dz dz 2i 2i 2 d cos(z) = − sin(z) 2. dz 1 d tan(z) = 3. dz cos2 (z) 1 d cot(z) = − 2 4. dz sin (z) 1. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Derivatives 1. 2. 3. 4. 5. d d eiz −e−iz ieiz +ie−iz eiz +e−iz sin(z) = = = = cos(z) dz dz 2i 2i 2 d cos(z) = − sin(z) dz 1 d tan(z) = dz cos2 (z) 1 d cot(z) = − 2 dz sin (z) d sec(z) = sec(z) tan(z) dz Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Derivatives 1. 2. 3. 4. 5. 6. d d eiz −e−iz ieiz +ie−iz eiz +e−iz sin(z) = = = = cos(z) dz dz 2i 2i 2 d cos(z) = − sin(z) dz 1 d tan(z) = dz cos2 (z) 1 d cot(z) = − 2 dz sin (z) d sec(z) = sec(z) tan(z) dz d csc(z) = − csc(z) cot(z) dz Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions The Derivatives d d eiz −e−iz ieiz +ie−iz eiz +e−iz sin(z) = = = = cos(z) dz dz 2i 2i 2 d cos(z) = − sin(z) 2. dz 1 d tan(z) = 3. dz cos2 (z) 1 d cot(z) = − 2 4. dz sin (z) d 5. sec(z) = sec(z) tan(z) dz d 6. csc(z) = − csc(z) cot(z) dz Note that secant and cosecant are not very common (around the world). 1. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Definition. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Definition. For any complex number z we define cosh(z) = Bernd Schröder Trigonometric and Hyperbolic Functions ez + e−z 2 logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Definition. For any complex number z we define cosh(z) = ez + e−z 2 sinh(z) = ez − e−z . 2 and Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Definition. For any complex number z we define cosh(z) = ez + e−z 2 sinh(z) = ez − e−z . 2 and Note. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Definition. For any complex number z we define cosh(z) = ez + e−z 2 sinh(z) = ez − e−z . 2 and Note. sin(iz) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Definition. For any complex number z we define cosh(z) = ez + e−z 2 sinh(z) = ez − e−z . 2 and Note. sin(iz) = eiiz − e−iiz 2i Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Definition. For any complex number z we define cosh(z) = ez + e−z 2 sinh(z) = ez − e−z . 2 and Note. sin(iz) = eiiz − e−iiz e−z − ez = 2i 2i Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Definition. For any complex number z we define cosh(z) = ez + e−z 2 sinh(z) = ez − e−z . 2 and Note. sin(iz) = eiiz − e−iiz e−z − ez ez − e−z = = ii 2i 2i 2i Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Definition. For any complex number z we define cosh(z) = ez + e−z 2 sinh(z) = ez − e−z . 2 and Note. sin(iz) = eiiz − e−iiz e−z − ez ez − e−z = = ii = i sinh(z) 2i 2i 2i Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Definition. For any complex number z we define cosh(z) = ez + e−z 2 sinh(z) = ez − e−z . 2 and Note. sin(iz) = cos(iz) eiiz − e−iiz e−z − ez ez − e−z = = ii = i sinh(z) 2i 2i 2i Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Definition. For any complex number z we define cosh(z) = ez + e−z 2 sinh(z) = ez − e−z . 2 and Note. eiiz − e−iiz e−z − ez ez − e−z = = ii = i sinh(z) 2i 2i 2i cos(iz) = cosh(z) sin(iz) = Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Another Connection Between Sine and Hyperbolic Sine Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Another Connection Between Sine and Hyperbolic Sine 2 | sin(z)| Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Another Connection Between Sine and Hyperbolic Sine 2 | sin(z)| = sin(x + iy)sin(x + iy) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Another Connection Between Sine and Hyperbolic Sine 2 | sin(z)| = sin(x + iy)sin(x + iy) = sin(x) cos(iy) + sin(iy) cos(x) sin(x + iy) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Another Connection Between Sine and Hyperbolic Sine 2 | sin(z)| = sin(x + iy)sin(x + iy) = = sin(x) cos(iy) + sin(iy) cos(x) sin(x + iy) sin(x) cosh(y) + i sinh(y) cos(x) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Another Connection Between Sine and Hyperbolic Sine 2 | sin(z)| = sin(x + iy)sin(x + iy) sin(x) cos(iy) + sin(iy) cos(x) sin(x + iy) = sin(x) cosh(y) + i sinh(y) cos(x) × × sin(x) cosh(y) − i sinh(y) cos(x) = Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Another Connection Between Sine and Hyperbolic Sine 2 | sin(z)| = sin(x + iy)sin(x + iy) sin(x) cos(iy) + sin(iy) cos(x) sin(x + iy) = sin(x) cosh(y) + i sinh(y) cos(x) × × sin(x) cosh(y) − i sinh(y) cos(x) = = sin2 (x) cosh2 (y) + sinh2 (y) cos2 (x) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Another Connection Between Sine and Hyperbolic Sine 2 | sin(z)| = sin(x + iy)sin(x + iy) sin(x) cos(iy) + sin(iy) cos(x) sin(x + iy) = sin(x) cosh(y) + i sinh(y) cos(x) × × sin(x) cosh(y) − i sinh(y) cos(x) = = sin2 (x) cosh2 (y) + sinh2 (y) cos2 (x) + sinh2 (y) sin2 (x) − sinh2 (y) sin2 (x) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Another Connection Between Sine and Hyperbolic Sine 2 | sin(z)| = sin(x + iy)sin(x + iy) sin(x) cos(iy) + sin(iy) cos(x) sin(x + iy) = sin(x) cosh(y) + i sinh(y) cos(x) × × sin(x) cosh(y) − i sinh(y) cos(x) = = sin2 (x) cosh2 (y) + sinh2 (y) cos2 (x) + sinh2 (y) sin2 (x) − sinh2 (y) sin2 (x) = sin2 (x) cosh2 (y) − sinh2 (y) + sinh2 (y) cos2 (x) + sin2 (x) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Another Connection Between Sine and Hyperbolic Sine 2 | sin(z)| = sin(x + iy)sin(x + iy) sin(x) cos(iy) + sin(iy) cos(x) sin(x + iy) = sin(x) cosh(y) + i sinh(y) cos(x) × × sin(x) cosh(y) − i sinh(y) cos(x) = = sin2 (x) cosh2 (y) + sinh2 (y) cos2 (x) + sinh2 (y) sin2 (x) − sinh2 (y) sin2 (x) = sin2 (x) cosh2 (y) − sinh2 (y) + sinh2 (y) cos2 (x) + sin2 (x) = sin2 (x) + sinh2 (y) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. The only zeros of the complex sine function are the numbers nπ where n is an integer. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. The only zeros of the complex sine function are the numbers nπ where n is an integer. Proof. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. The only zeros of the complex sine function are the numbers nπ where n is an integer. Proof. The only solutions of 2 0 = sin(z) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. The only zeros of the complex sine function are the numbers nπ where n is an integer. Proof. The only solutions of 2 2 0 = sin(z) = sin(x + iy) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. The only zeros of the complex sine function are the numbers nπ where n is an integer. Proof. The only solutions of 2 2 0 = sin(z) = sin(x + iy) = sin2 (x) + sinh2 (y) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. The only zeros of the complex sine function are the numbers nπ where n is an integer. Proof. The only solutions of 2 2 0 = sin(z) = sin(x + iy) = sin2 (x) + sinh2 (y) are y = 0 Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. The only zeros of the complex sine function are the numbers nπ where n is an integer. Proof. The only solutions of 2 2 0 = sin(z) = sin(x + iy) = sin2 (x) + sinh2 (y) are y = 0 and x = nπ. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. The only zeros of the complex sine function are the numbers nπ where n is an integer. Proof. The only solutions of 2 2 0 = sin(z) = sin(x + iy) = sin2 (x) + sinh2 (y) are y = 0 and x = nπ. So z = nπ where n is an integer are the zeros of the sine function. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. The only zeros of the complex sine function are the numbers nπ where n is an integer. Proof. The only solutions of 2 2 0 = sin(z) = sin(x + iy) = sin2 (x) + sinh2 (y) are y = 0 and x = nπ. So z = nπ where n is an integer are the zeros of the sine function. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Identities for Hyperbolic Functions Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Identities for Hyperbolic Functions 1. d sinh(z) = cosh(z) dz Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Identities for Hyperbolic Functions d sinh(z) = cosh(z) dz d cosh(z) = sinh(z) 2. dz 1. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Identities for Hyperbolic Functions d sinh(z) = cosh(z) dz d cosh(z) = sinh(z) 2. dz 3. sinh(z1 + z2 ) = sinh(z1 ) cosh(z2 ) + cosh(z1 ) sinh(z2 ) 1. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Identities for Hyperbolic Functions d sinh(z) = cosh(z) dz d cosh(z) = sinh(z) 2. dz 3. sinh(z1 + z2 ) = sinh(z1 ) cosh(z2 ) + cosh(z1 ) sinh(z2 ) 4. cosh(z1 + z2 ) = cosh(z1 ) cosh(z2 ) + sinh(z1 ) sinh(z2 ) 1. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Identities for Hyperbolic Functions 1. 2. 3. 4. 5. d sinh(z) = cosh(z) dz d cosh(z) = sinh(z) dz sinh(z1 + z2 ) = sinh(z1 ) cosh(z2 ) + cosh(z1 ) sinh(z2 ) cosh(z1 + z2 ) = cosh(z1 ) cosh(z2 ) + sinh(z1 ) sinh(z2 ) cosh2 (z) − sinh2 (z) = 1 Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Identities for Hyperbolic Functions 1. 2. 3. 4. 5. 6. d sinh(z) = cosh(z) dz d cosh(z) = sinh(z) dz sinh(z1 + z2 ) = sinh(z1 ) cosh(z2 ) + cosh(z1 ) sinh(z2 ) cosh(z1 + z2 ) = cosh(z1 ) cosh(z2 ) + sinh(z1 ) sinh(z2 ) cosh2 (z) − sinh2 (z) = 1 sinh(z + 2πi) = sinh(z) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Identities for Hyperbolic Functions 1. 2. 3. 4. 5. 6. 7. d sinh(z) = cosh(z) dz d cosh(z) = sinh(z) dz sinh(z1 + z2 ) = sinh(z1 ) cosh(z2 ) + cosh(z1 ) sinh(z2 ) cosh(z1 + z2 ) = cosh(z1 ) cosh(z2 ) + sinh(z1 ) sinh(z2 ) cosh2 (z) − sinh2 (z) = 1 sinh(z + 2πi) = sinh(z) cosh(z + 2πi) = cosh(z) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Identities for Hyperbolic Functions d sinh(z) = cosh(z) dz d cosh(z) = sinh(z) 2. dz 3. sinh(z1 + z2 ) = sinh(z1 ) cosh(z2 ) + cosh(z1 ) sinh(z2 ) 4. cosh(z1 + z2 ) = cosh(z1 ) cosh(z2 ) + sinh(z1 ) sinh(z2 ) 5. cosh2 (z) − sinh2 (z) = 1 6. sinh(z + 2πi) = sinh(z) 7. cosh(z + 2πi) = cosh(z) ... and more. 1. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Identities for Hyperbolic Functions d sinh(z) = cosh(z) dz d cosh(z) = sinh(z) 2. dz 3. sinh(z1 + z2 ) = sinh(z1 ) cosh(z2 ) + cosh(z1 ) sinh(z2 ) 4. cosh(z1 + z2 ) = cosh(z1 ) cosh(z2 ) + sinh(z1 ) sinh(z2 ) 5. cosh2 (z) − sinh2 (z) = 1 6. sinh(z + 2πi) = sinh(z) 7. cosh(z + 2πi) = cosh(z) ... and more. And they can all be verified straight from the definition. 1. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. The only zeros of the complex hyperbolic sine function are the numbers nπi, where n is an integer. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. The only zeros of the complex hyperbolic sine function are the numbers nπi, where n is an integer. Proof. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. The only zeros of the complex hyperbolic sine function are the numbers nπi, where n is an integer. Proof. sinh(z) = −i sin(iz) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. The only zeros of the complex hyperbolic sine function are the numbers nπi, where n is an integer. Proof. sinh(z) = −i sin(iz) is equal to zero where sin(iz) is equal to zero. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. The only zeros of the complex hyperbolic sine function are the numbers nπi, where n is an integer. Proof. sinh(z) = −i sin(iz) is equal to zero where sin(iz) is equal to zero. Hence the zeros of sinh(z) are at z = nπi, where n is an integer. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. The only zeros of the complex hyperbolic sine function are the numbers nπi, where n is an integer. Proof. sinh(z) = −i sin(iz) is equal to zero where sin(iz) is equal to zero. Hence the zeros of sinh(z) are at z = nπi, where n is an integer. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions More Hyperbolic Functions Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions More Hyperbolic Functions 1. tanh(z) := sinh(z) cosh(z) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions More Hyperbolic Functions sinh(z) cosh(z) cosh(z) 2. coth(z) := sinh(z) 1. tanh(z) := Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions More Hyperbolic Functions sinh(z) cosh(z) cosh(z) 2. coth(z) := sinh(z) 1 3. sech(z) := cosh(z) 1. tanh(z) := Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions More Hyperbolic Functions sinh(z) cosh(z) cosh(z) 2. coth(z) := sinh(z) 1 3. sech(z) := cosh(z) 1 4. csch(z) := sinh(z) 1. tanh(z) := Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions More Hyperbolic Functions sinh(z) cosh(z) cosh(z) 2. coth(z) := sinh(z) 1 3. sech(z) := cosh(z) 1 4. csch(z) := sinh(z) Note that hyperbolic secant and cosecant are not very common (around the world) either. 1. tanh(z) := Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Remaining Derivatives Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Remaining Derivatives 1. d 1 tanh(z) = dz cosh2 (z) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Remaining Derivatives d 1 tanh(z) = dz cosh2 (z) 1 d coth(z) = − 2. dz sinh2 (z) 1. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Remaining Derivatives d 1 tanh(z) = dz cosh2 (z) 1 d coth(z) = − 2. dz sinh2 (z) d 3. sech(z) = −sech(z) tanh(z) dz 1. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Remaining Derivatives d 1 tanh(z) = dz cosh2 (z) 1 d coth(z) = − 2. dz sinh2 (z) d 3. sech(z) = −sech(z) tanh(z) dz d 4. csch(z) = −csch(z) coth(z) dz 1. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Remaining Derivatives d 1 tanh(z) = dz cosh2 (z) 1 d coth(z) = − 2. dz sinh2 (z) d 3. sech(z) = −sech(z) tanh(z) dz d 4. csch(z) = −csch(z) coth(z) dz Note that hyperbolic secant and cosecant are not very common (around the world) either. 1. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. For all complex numbers z we have 1 2 2 arccos(z) = −i log z + i 1 − z Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. For all complex numbers z we have 1 2 2 arccos(z) = −i log z + i 1 − z where the right side is a multivalued function. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Proof. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Proof. w = arccos(z) Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Proof. w = arccos(z) cos(w) = z Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Proof. w = arccos(z) cos(w) = z eiw + e−iw 2 Bernd Schröder Trigonometric and Hyperbolic Functions = z logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Proof. w = arccos(z) cos(w) = z eiw + e−iw = z 2 eiw − 2z + e−iw = 0 Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Proof. w = arccos(z) cos(w) = z eiw + e−iw = z 2 eiw − 2z + e−iw = 0 2 eiw − 2zeiw + 1 = 0 Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Proof. w = arccos(z) cos(w) = z eiw + e−iw = z 2 eiw − 2z + e−iw = 0 2 eiw − 2zeiw + 1 = 0 iw e Bernd Schröder Trigonometric and Hyperbolic Functions = √ 2z ± 4z2 − 4 2 logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Proof. w = arccos(z) cos(w) = z eiw + e−iw = z 2 eiw − 2z + e−iw = 0 2 eiw − 2zeiw + 1 = 0 iw e Bernd Schröder Trigonometric and Hyperbolic Functions √ 2z ± 4z2 − 4 = p2 = z ± z2 − 1 logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Proof. w = arccos(z) cos(w) = z eiw + e−iw = z 2 eiw − 2z + e−iw = 0 2 eiw − 2zeiw + 1 = 0 √ 2z ± 4z2 − 4 e = p2 = z ± z2 − 1 p 1 2 w = log z ± z − 1 i iw Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Proof. w = arccos(z) cos(w) = z eiw + e−iw = z 2 eiw − 2z + e−iw = 0 2 eiw − 2zeiw + 1 = 0 iw e = = w = = Bernd Schröder Trigonometric and Hyperbolic Functions √ 2z ± 4z2 − 4 p2 z ± z2 − 1 p 1 2 log z ± z − 1 i p −i log z ± i 1 − z2 logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Proof. w = arccos(z) cos(w) = z eiw + e−iw = z 2 eiw − 2z + e−iw = 0 2 eiw − 2zeiw + 1 = 0 iw e = = w = = Bernd Schröder Trigonometric and Hyperbolic Functions √ 2z ± 4z2 − 4 p2 z ± z2 − 1 p 1 2 log z ± z − 1 i p −i log z ± i 1 − z2 logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Similarly We Can Derive ... Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Similarly We Can Derive ... 1. arcsin(z) = −i log iz + 1 − z Bernd Schröder Trigonometric and Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions 2 12 logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Similarly We Can Derive ... 1. arcsin(z) = −i log iz + 1 − z i i+z 2. arctan(z) = log 2 i−z Bernd Schröder Trigonometric and Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions 2 12 logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Similarly We Can Derive ... Inverse Trigonometric and Hyperbolic Functions 2 12 1. arcsin(z) = −i log iz + 1 − z i i+z 2. arctan(z) = log 2 i−z 1 2 2 3. arsinh(z) = log z + 1 + z Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Similarly We Can Derive ... Inverse Trigonometric and Hyperbolic Functions 2 12 1. arcsin(z) = −i log iz + 1 − z i i+z 2. arctan(z) = log 2 i−z 1 2 2 3. arsinh(z) = log z + 1 + z 21 2 4. arcosh(z) = log z + z − 1 Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Similarly We Can Derive ... Inverse Trigonometric and Hyperbolic Functions 2 12 1. arcsin(z) = −i log iz + 1 − z i i+z 2. arctan(z) = log 2 i−z 1 2 2 3. arsinh(z) = log z + 1 + z 21 2 4. arcosh(z) = log z + z − 1 1 1+z 5. artanh(z) = log 2 1−z Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Theorem. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Theorem. Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions d 1 arcsin(z) = 1 dz (1 − z2 ) 2 Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions d 1 arcsin(z) = 1 where the right side is dz (1 − z2 ) 2 multivalued again. Theorem. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions d 1 arcsin(z) = 1 where the right side is dz (1 − z2 ) 2 multivalued again. Theorem. Proof. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions d 1 arcsin(z) = 1 where the right side is dz (1 − z2 ) 2 multivalued again. Theorem. Proof. d arcsin(z) dz Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions d 1 arcsin(z) = 1 where the right side is dz (1 − z2 ) 2 multivalued again. Theorem. Proof. d arcsin(z) = dz Bernd Schröder Trigonometric and Hyperbolic Functions 1 d 2 2 −i log iz + 1 − z dz logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions d 1 arcsin(z) = 1 where the right side is dz (1 − z2 ) 2 multivalued again. Theorem. Proof. d arcsin(z) = dz 1 d 2 2 −i log iz + 1 − z dz = −i 1 1 iz + (1 − z2 ) 2 Bernd Schröder Trigonometric and Hyperbolic Functions ! 1 1 i+ (−2z) 2 (1 − z2 ) 12 logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions d 1 arcsin(z) = 1 where the right side is dz (1 − z2 ) 2 multivalued again. Theorem. Proof. d arcsin(z) = dz 1 d 2 2 −i log iz + 1 − z dz ! 1 1 = −i i+ (−2z) 1 2 (1 − z2 ) 12 iz + (1 − z2 ) 2 1 i 1 − z2 2 − z −i = 1 1 iz + (1 − z2 ) 2 (1 − z2 ) 2 1 Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions d 1 arcsin(z) = 1 where the right side is dz (1 − z2 ) 2 multivalued again. Theorem. Proof. d arcsin(z) = dz 1 d 2 2 −i log iz + 1 − z dz ! 1 1 = −i i+ (−2z) 1 2 (1 − z2 ) 12 iz + (1 − z2 ) 2 1 i 1 − z2 2 − z −i = 1 1 iz + (1 − z2 ) 2 (1 − z2 ) 2 1 = 1 1 (1 − z2 ) 2 Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions d 1 arcsin(z) = 1 where the right side is dz (1 − z2 ) 2 multivalued again. Theorem. Proof. d arcsin(z) = dz 1 d 2 2 −i log iz + 1 − z dz ! 1 1 = −i i+ (−2z) 1 2 (1 − z2 ) 12 iz + (1 − z2 ) 2 1 i 1 − z2 2 − z −i = 1 1 iz + (1 − z2 ) 2 (1 − z2 ) 2 1 = 1 1 (1 − z2 ) 2 Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Similarly ... Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Similarly ... 1. d 1 arccos(z) = − 1 dz (1 − z2 ) 2 Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions Similarly ... d 1 arccos(z) = − 1 dz (1 − z2 ) 2 1 d arctan(z) = 2. dz 1 + z2 1. Bernd Schröder Trigonometric and Hyperbolic Functions logo1 Louisiana Tech University, College of Engineering and Science
