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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | efval2 15901* | Value of the exponential function. (Contributed by Mario Carneiro, 29-Apr-2014.) |
β’ πΉ = (π β β0 β¦ ((π΄βπ) / (!βπ))) β β’ (π΄ β β β (expβπ΄) = Ξ£π β β0 (πΉβπ)) | ||
Theorem | efcvg 15902* | The series that defines the exponential function converges to it. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.) |
β’ πΉ = (π β β0 β¦ ((π΄βπ) / (!βπ))) β β’ (π΄ β β β seq0( + , πΉ) β (expβπ΄)) | ||
Theorem | efcvgfsum 15903* | Exponential function convergence in terms of a sequence of partial finite sums. (Contributed by NM, 10-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.) |
β’ πΉ = (π β β0 β¦ Ξ£π β (0...π)((π΄βπ) / (!βπ))) β β’ (π΄ β β β πΉ β (expβπ΄)) | ||
Theorem | reefcl 15904 | The exponential function is real if its argument is real. (Contributed by NM, 27-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.) |
β’ (π΄ β β β (expβπ΄) β β) | ||
Theorem | reefcld 15905 | The exponential function is real if its argument is real. (Contributed by Mario Carneiro, 29-May-2016.) |
β’ (π β π΄ β β) β β’ (π β (expβπ΄) β β) | ||
Theorem | ere 15906 | Euler's constant e = 2.71828... is a real number. (Contributed by NM, 19-Mar-2005.) (Revised by Steve Rodriguez, 8-Mar-2006.) |
β’ e β β | ||
Theorem | ege2le3 15907 | Lemma for egt2lt3 16023. (Contributed by NM, 20-Mar-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.) |
β’ πΉ = (π β β β¦ (2 Β· ((1 / 2)βπ))) & β’ πΊ = (π β β0 β¦ (1 / (!βπ))) β β’ (2 β€ e β§ e β€ 3) | ||
Theorem | ef0 15908 | Value of the exponential function at 0. Equation 2 of [Gleason] p. 308. (Contributed by Steve Rodriguez, 27-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.) |
β’ (expβ0) = 1 | ||
Theorem | efcj 15909 | The exponential of a complex conjugate. Equation 3 of [Gleason] p. 308. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.) |
β’ (π΄ β β β (expβ(ββπ΄)) = (ββ(expβπ΄))) | ||
Theorem | efaddlem 15910* | Lemma for efadd 15911 (exponential function addition law). (Contributed by Mario Carneiro, 29-Apr-2014.) |
β’ πΉ = (π β β0 β¦ ((π΄βπ) / (!βπ))) & β’ πΊ = (π β β0 β¦ ((π΅βπ) / (!βπ))) & β’ π» = (π β β0 β¦ (((π΄ + π΅)βπ) / (!βπ))) & β’ (π β π΄ β β) & β’ (π β π΅ β β) β β’ (π β (expβ(π΄ + π΅)) = ((expβπ΄) Β· (expβπ΅))) | ||
Theorem | efadd 15911 | Sum of exponents law for exponential function. (Contributed by NM, 10-Jan-2006.) (Proof shortened by Mario Carneiro, 29-Apr-2014.) |
β’ ((π΄ β β β§ π΅ β β) β (expβ(π΄ + π΅)) = ((expβπ΄) Β· (expβπ΅))) | ||
Theorem | fprodefsum 15912* | Move the exponential function from inside a finite product to outside a finite sum. (Contributed by Scott Fenton, 26-Dec-2017.) |
β’ π = (β€β₯βπ) & β’ (π β π β π) & β’ ((π β§ π β π) β π΄ β β) β β’ (π β βπ β (π...π)(expβπ΄) = (expβΞ£π β (π...π)π΄)) | ||
Theorem | efcan 15913 | Cancellation law for exponential function. Equation 27 of [Rudin] p. 164. (Contributed by NM, 13-Jan-2006.) |
β’ (π΄ β β β ((expβπ΄) Β· (expβ-π΄)) = 1) | ||
Theorem | efne0 15914 | The exponential of a complex number is nonzero. Corollary 15-4.3 of [Gleason] p. 309. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 29-Apr-2014.) |
β’ (π΄ β β β (expβπ΄) β 0) | ||
Theorem | efneg 15915 | The exponential of the opposite is the inverse of the exponential. (Contributed by Mario Carneiro, 10-May-2014.) |
β’ (π΄ β β β (expβ-π΄) = (1 / (expβπ΄))) | ||
Theorem | eff2 15916 | The exponential function maps the complex numbers to the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.) |
β’ exp:ββΆ(β β {0}) | ||
Theorem | efsub 15917 | Difference of exponents law for exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.) |
β’ ((π΄ β β β§ π΅ β β) β (expβ(π΄ β π΅)) = ((expβπ΄) / (expβπ΅))) | ||
Theorem | efexp 15918 | The exponential of an integer power. Corollary 15-4.4 of [Gleason] p. 309, restricted to integers. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 5-Jun-2014.) |
β’ ((π΄ β β β§ π β β€) β (expβ(π Β· π΄)) = ((expβπ΄)βπ)) | ||
Theorem | efzval 15919 | Value of the exponential function for integers. Special case of efval 15897. Equation 30 of [Rudin] p. 164. (Contributed by Steve Rodriguez, 15-Sep-2006.) (Revised by Mario Carneiro, 5-Jun-2014.) |
β’ (π β β€ β (expβπ) = (eβπ)) | ||
Theorem | efgt0 15920 | The exponential of a real number is greater than 0. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ (π΄ β β β 0 < (expβπ΄)) | ||
Theorem | rpefcl 15921 | The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 10-Nov-2013.) |
β’ (π΄ β β β (expβπ΄) β β+) | ||
Theorem | rpefcld 15922 | The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.) |
β’ (π β π΄ β β) β β’ (π β (expβπ΄) β β+) | ||
Theorem | eftlcvg 15923* | The tail series of the exponential function are convergent. (Contributed by Mario Carneiro, 29-Apr-2014.) |
β’ πΉ = (π β β0 β¦ ((π΄βπ) / (!βπ))) β β’ ((π΄ β β β§ π β β0) β seqπ( + , πΉ) β dom β ) | ||
Theorem | eftlcl 15924* | Closure of the sum of an infinite tail of the series defining the exponential function. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ πΉ = (π β β0 β¦ ((π΄βπ) / (!βπ))) β β’ ((π΄ β β β§ π β β0) β Ξ£π β (β€β₯βπ)(πΉβπ) β β) | ||
Theorem | reeftlcl 15925* | Closure of the sum of an infinite tail of the series defining the exponential function. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ πΉ = (π β β0 β¦ ((π΄βπ) / (!βπ))) β β’ ((π΄ β β β§ π β β0) β Ξ£π β (β€β₯βπ)(πΉβπ) β β) | ||
Theorem | eftlub 15926* | An upper bound on the absolute value of the infinite tail of the series expansion of the exponential function on the closed unit disk. (Contributed by Paul Chapman, 19-Jan-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.) |
β’ πΉ = (π β β0 β¦ ((π΄βπ) / (!βπ))) & β’ πΊ = (π β β0 β¦ (((absβπ΄)βπ) / (!βπ))) & β’ π» = (π β β0 β¦ ((((absβπ΄)βπ) / (!βπ)) Β· ((1 / (π + 1))βπ))) & β’ (π β π β β) & β’ (π β π΄ β β) & β’ (π β (absβπ΄) β€ 1) β β’ (π β (absβΞ£π β (β€β₯βπ)(πΉβπ)) β€ (((absβπ΄)βπ) Β· ((π + 1) / ((!βπ) Β· π)))) | ||
Theorem | efsep 15927* | Separate out the next term of the power series expansion of the exponential function. The last hypothesis allows the separated terms to be rearranged as desired. (Contributed by Paul Chapman, 23-Nov-2007.) (Revised by Mario Carneiro, 29-Apr-2014.) |
β’ πΉ = (π β β0 β¦ ((π΄βπ) / (!βπ))) & β’ π = (π + 1) & β’ π β β0 & β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β (expβπ΄) = (π΅ + Ξ£π β (β€β₯βπ)(πΉβπ))) & β’ (π β (π΅ + ((π΄βπ) / (!βπ))) = π·) β β’ (π β (expβπ΄) = (π· + Ξ£π β (β€β₯βπ)(πΉβπ))) | ||
Theorem | effsumlt 15928* | The partial sums of the series expansion of the exponential function at a positive real number are bounded by the value of the function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 29-Apr-2014.) |
β’ πΉ = (π β β0 β¦ ((π΄βπ) / (!βπ))) & β’ (π β π΄ β β+) & β’ (π β π β β0) β β’ (π β (seq0( + , πΉ)βπ) < (expβπ΄)) | ||
Theorem | eft0val 15929 | The value of the first term of the series expansion of the exponential function is 1. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 29-Apr-2014.) |
β’ (π΄ β β β ((π΄β0) / (!β0)) = 1) | ||
Theorem | ef4p 15930* | Separate out the first four terms of the infinite series expansion of the exponential function. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.) |
β’ πΉ = (π β β0 β¦ ((π΄βπ) / (!βπ))) β β’ (π΄ β β β (expβπ΄) = ((((1 + π΄) + ((π΄β2) / 2)) + ((π΄β3) / 6)) + Ξ£π β (β€β₯β4)(πΉβπ))) | ||
Theorem | efgt1p2 15931 | The exponential of a positive real number is greater than the sum of the first three terms of the series expansion. (Contributed by Mario Carneiro, 15-Sep-2014.) |
β’ (π΄ β β+ β ((1 + π΄) + ((π΄β2) / 2)) < (expβπ΄)) | ||
Theorem | efgt1p 15932 | The exponential of a positive real number is greater than 1 plus that number. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ (π΄ β β+ β (1 + π΄) < (expβπ΄)) | ||
Theorem | efgt1 15933 | The exponential of a positive real number is greater than 1. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ (π΄ β β+ β 1 < (expβπ΄)) | ||
Theorem | eflt 15934 | The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 17-Jul-2014.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ < π΅ β (expβπ΄) < (expβπ΅))) | ||
Theorem | efle 15935 | The exponential function on the reals is nondecreasing. (Contributed by Mario Carneiro, 11-Mar-2014.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ β€ π΅ β (expβπ΄) β€ (expβπ΅))) | ||
Theorem | reef11 15936 | The exponential function on real numbers is one-to-one. (Contributed by NM, 21-Aug-2008.) (Revised by Mario Carneiro, 11-Mar-2014.) |
β’ ((π΄ β β β§ π΅ β β) β ((expβπ΄) = (expβπ΅) β π΄ = π΅)) | ||
Theorem | reeff1 15937 | The exponential function maps real arguments one-to-one to positive reals. (Contributed by Steve Rodriguez, 25-Aug-2007.) (Revised by Mario Carneiro, 10-Nov-2013.) |
β’ (exp βΎ β):ββ1-1ββ+ | ||
Theorem | eflegeo 15938 | The exponential function on the reals between 0 and 1 lies below the comparable geometric series sum. (Contributed by Paul Chapman, 11-Sep-2007.) |
β’ (π β π΄ β β) & β’ (π β 0 β€ π΄) & β’ (π β π΄ < 1) β β’ (π β (expβπ΄) β€ (1 / (1 β π΄))) | ||
Theorem | sinval 15939 | Value of the sine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.) |
β’ (π΄ β β β (sinβπ΄) = (((expβ(i Β· π΄)) β (expβ(-i Β· π΄))) / (2 Β· i))) | ||
Theorem | cosval 15940 | Value of the cosine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.) |
β’ (π΄ β β β (cosβπ΄) = (((expβ(i Β· π΄)) + (expβ(-i Β· π΄))) / 2)) | ||
Theorem | sinf 15941 | Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ sin:ββΆβ | ||
Theorem | cosf 15942 | Domain and codomain of the cosine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ cos:ββΆβ | ||
Theorem | sincl 15943 | Closure of the sine function. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ (π΄ β β β (sinβπ΄) β β) | ||
Theorem | coscl 15944 | Closure of the cosine function with a complex argument. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ (π΄ β β β (cosβπ΄) β β) | ||
Theorem | tanval 15945 | Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.) |
β’ ((π΄ β β β§ (cosβπ΄) β 0) β (tanβπ΄) = ((sinβπ΄) / (cosβπ΄))) | ||
Theorem | tancl 15946 | The closure of the tangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.) |
β’ ((π΄ β β β§ (cosβπ΄) β 0) β (tanβπ΄) β β) | ||
Theorem | sincld 15947 | Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.) |
β’ (π β π΄ β β) β β’ (π β (sinβπ΄) β β) | ||
Theorem | coscld 15948 | Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.) |
β’ (π β π΄ β β) β β’ (π β (cosβπ΄) β β) | ||
Theorem | tancld 15949 | Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) |
β’ (π β π΄ β β) & β’ (π β (cosβπ΄) β 0) β β’ (π β (tanβπ΄) β β) | ||
Theorem | tanval2 15950 | Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) |
β’ ((π΄ β β β§ (cosβπ΄) β 0) β (tanβπ΄) = (((expβ(i Β· π΄)) β (expβ(-i Β· π΄))) / (i Β· ((expβ(i Β· π΄)) + (expβ(-i Β· π΄)))))) | ||
Theorem | tanval3 15951 | Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) |
β’ ((π΄ β β β§ ((expβ(2 Β· (i Β· π΄))) + 1) β 0) β (tanβπ΄) = (((expβ(2 Β· (i Β· π΄))) β 1) / (i Β· ((expβ(2 Β· (i Β· π΄))) + 1)))) | ||
Theorem | resinval 15952 | The sine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.) |
β’ (π΄ β β β (sinβπ΄) = (ββ(expβ(i Β· π΄)))) | ||
Theorem | recosval 15953 | The cosine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.) |
β’ (π΄ β β β (cosβπ΄) = (ββ(expβ(i Β· π΄)))) | ||
Theorem | efi4p 15954* | Separate out the first four terms of the infinite series expansion of the exponential function. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ πΉ = (π β β0 β¦ (((i Β· π΄)βπ) / (!βπ))) β β’ (π΄ β β β (expβ(i Β· π΄)) = (((1 β ((π΄β2) / 2)) + (i Β· (π΄ β ((π΄β3) / 6)))) + Ξ£π β (β€β₯β4)(πΉβπ))) | ||
Theorem | resin4p 15955* | Separate out the first four terms of the infinite series expansion of the sine of a real number. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ πΉ = (π β β0 β¦ (((i Β· π΄)βπ) / (!βπ))) β β’ (π΄ β β β (sinβπ΄) = ((π΄ β ((π΄β3) / 6)) + (ββΞ£π β (β€β₯β4)(πΉβπ)))) | ||
Theorem | recos4p 15956* | Separate out the first four terms of the infinite series expansion of the cosine of a real number. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ πΉ = (π β β0 β¦ (((i Β· π΄)βπ) / (!βπ))) β β’ (π΄ β β β (cosβπ΄) = ((1 β ((π΄β2) / 2)) + (ββΞ£π β (β€β₯β4)(πΉβπ)))) | ||
Theorem | resincl 15957 | The sine of a real number is real. (Contributed by NM, 30-Apr-2005.) |
β’ (π΄ β β β (sinβπ΄) β β) | ||
Theorem | recoscl 15958 | The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.) |
β’ (π΄ β β β (cosβπ΄) β β) | ||
Theorem | retancl 15959 | The closure of the tangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.) |
β’ ((π΄ β β β§ (cosβπ΄) β 0) β (tanβπ΄) β β) | ||
Theorem | resincld 15960 | Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.) |
β’ (π β π΄ β β) β β’ (π β (sinβπ΄) β β) | ||
Theorem | recoscld 15961 | Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.) |
β’ (π β π΄ β β) β β’ (π β (cosβπ΄) β β) | ||
Theorem | retancld 15962 | Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) |
β’ (π β π΄ β β) & β’ (π β (cosβπ΄) β 0) β β’ (π β (tanβπ΄) β β) | ||
Theorem | sinneg 15963 | The sine of a negative is the negative of the sine. (Contributed by NM, 30-Apr-2005.) |
β’ (π΄ β β β (sinβ-π΄) = -(sinβπ΄)) | ||
Theorem | cosneg 15964 | The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.) |
β’ (π΄ β β β (cosβ-π΄) = (cosβπ΄)) | ||
Theorem | tanneg 15965 | The tangent of a negative is the negative of the tangent. (Contributed by David A. Wheeler, 23-Mar-2014.) |
β’ ((π΄ β β β§ (cosβπ΄) β 0) β (tanβ-π΄) = -(tanβπ΄)) | ||
Theorem | sin0 15966 | Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.) |
β’ (sinβ0) = 0 | ||
Theorem | cos0 15967 | Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.) |
β’ (cosβ0) = 1 | ||
Theorem | tan0 15968 | The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.) |
β’ (tanβ0) = 0 | ||
Theorem | efival 15969 | The exponential function in terms of sine and cosine. (Contributed by NM, 30-Apr-2005.) |
β’ (π΄ β β β (expβ(i Β· π΄)) = ((cosβπ΄) + (i Β· (sinβπ΄)))) | ||
Theorem | efmival 15970 | The exponential function in terms of sine and cosine. (Contributed by NM, 14-Jan-2006.) |
β’ (π΄ β β β (expβ(-i Β· π΄)) = ((cosβπ΄) β (i Β· (sinβπ΄)))) | ||
Theorem | sinhval 15971 | Value of the hyperbolic sine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.) |
β’ (π΄ β β β ((sinβ(i Β· π΄)) / i) = (((expβπ΄) β (expβ-π΄)) / 2)) | ||
Theorem | coshval 15972 | Value of the hyperbolic cosine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.) |
β’ (π΄ β β β (cosβ(i Β· π΄)) = (((expβπ΄) + (expβ-π΄)) / 2)) | ||
Theorem | resinhcl 15973 | The hyperbolic sine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.) |
β’ (π΄ β β β ((sinβ(i Β· π΄)) / i) β β) | ||
Theorem | rpcoshcl 15974 | The hyperbolic cosine of a real number is a positive real. (Contributed by Mario Carneiro, 4-Apr-2015.) |
β’ (π΄ β β β (cosβ(i Β· π΄)) β β+) | ||
Theorem | recoshcl 15975 | The hyperbolic cosine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.) |
β’ (π΄ β β β (cosβ(i Β· π΄)) β β) | ||
Theorem | retanhcl 15976 | The hyperbolic tangent of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.) |
β’ (π΄ β β β ((tanβ(i Β· π΄)) / i) β β) | ||
Theorem | tanhlt1 15977 | The hyperbolic tangent of a real number is upper bounded by 1. (Contributed by Mario Carneiro, 4-Apr-2015.) |
β’ (π΄ β β β ((tanβ(i Β· π΄)) / i) < 1) | ||
Theorem | tanhbnd 15978 | The hyperbolic tangent of a real number is bounded by 1. (Contributed by Mario Carneiro, 4-Apr-2015.) |
β’ (π΄ β β β ((tanβ(i Β· π΄)) / i) β (-1(,)1)) | ||
Theorem | efeul 15979 | Eulerian representation of the complex exponential. (Suggested by Jeff Hankins, 3-Jul-2006.) (Contributed by NM, 4-Jul-2006.) |
β’ (π΄ β β β (expβπ΄) = ((expβ(ββπ΄)) Β· ((cosβ(ββπ΄)) + (i Β· (sinβ(ββπ΄)))))) | ||
Theorem | efieq 15980 | The exponentials of two imaginary numbers are equal iff their sine and cosine components are equal. (Contributed by Paul Chapman, 15-Mar-2008.) |
β’ ((π΄ β β β§ π΅ β β) β ((expβ(i Β· π΄)) = (expβ(i Β· π΅)) β ((cosβπ΄) = (cosβπ΅) β§ (sinβπ΄) = (sinβπ΅)))) | ||
Theorem | sinadd 15981 | Addition formula for sine. Equation 14 of [Gleason] p. 310. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ ((π΄ β β β§ π΅ β β) β (sinβ(π΄ + π΅)) = (((sinβπ΄) Β· (cosβπ΅)) + ((cosβπ΄) Β· (sinβπ΅)))) | ||
Theorem | cosadd 15982 | Addition formula for cosine. Equation 15 of [Gleason] p. 310. (Contributed by NM, 15-Jan-2006.) (Revised by Mario Carneiro, 30-Apr-2014.) |
β’ ((π΄ β β β§ π΅ β β) β (cosβ(π΄ + π΅)) = (((cosβπ΄) Β· (cosβπ΅)) β ((sinβπ΄) Β· (sinβπ΅)))) | ||
Theorem | tanaddlem 15983 | A useful intermediate step in tanadd 15984 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.) |
β’ (((π΄ β β β§ π΅ β β) β§ ((cosβπ΄) β 0 β§ (cosβπ΅) β 0)) β ((cosβ(π΄ + π΅)) β 0 β ((tanβπ΄) Β· (tanβπ΅)) β 1)) | ||
Theorem | tanadd 15984 | Addition formula for tangent. (Contributed by Mario Carneiro, 4-Apr-2015.) |
β’ (((π΄ β β β§ π΅ β β) β§ ((cosβπ΄) β 0 β§ (cosβπ΅) β 0 β§ (cosβ(π΄ + π΅)) β 0)) β (tanβ(π΄ + π΅)) = (((tanβπ΄) + (tanβπ΅)) / (1 β ((tanβπ΄) Β· (tanβπ΅))))) | ||
Theorem | sinsub 15985 | Sine of difference. (Contributed by Paul Chapman, 12-Oct-2007.) |
β’ ((π΄ β β β§ π΅ β β) β (sinβ(π΄ β π΅)) = (((sinβπ΄) Β· (cosβπ΅)) β ((cosβπ΄) Β· (sinβπ΅)))) | ||
Theorem | cossub 15986 | Cosine of difference. (Contributed by Paul Chapman, 12-Oct-2007.) |
β’ ((π΄ β β β§ π΅ β β) β (cosβ(π΄ β π΅)) = (((cosβπ΄) Β· (cosβπ΅)) + ((sinβπ΄) Β· (sinβπ΅)))) | ||
Theorem | addsin 15987 | Sum of sines. (Contributed by Paul Chapman, 12-Oct-2007.) |
β’ ((π΄ β β β§ π΅ β β) β ((sinβπ΄) + (sinβπ΅)) = (2 Β· ((sinβ((π΄ + π΅) / 2)) Β· (cosβ((π΄ β π΅) / 2))))) | ||
Theorem | subsin 15988 | Difference of sines. (Contributed by Paul Chapman, 12-Oct-2007.) |
β’ ((π΄ β β β§ π΅ β β) β ((sinβπ΄) β (sinβπ΅)) = (2 Β· ((cosβ((π΄ + π΅) / 2)) Β· (sinβ((π΄ β π΅) / 2))))) | ||
Theorem | sinmul 15989 | Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 15982 and cossub 15986. (Contributed by David A. Wheeler, 26-May-2015.) |
β’ ((π΄ β β β§ π΅ β β) β ((sinβπ΄) Β· (sinβπ΅)) = (((cosβ(π΄ β π΅)) β (cosβ(π΄ + π΅))) / 2)) | ||
Theorem | cosmul 15990 | Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 15982 and cossub 15986. (Contributed by David A. Wheeler, 26-May-2015.) |
β’ ((π΄ β β β§ π΅ β β) β ((cosβπ΄) Β· (cosβπ΅)) = (((cosβ(π΄ β π΅)) + (cosβ(π΄ + π΅))) / 2)) | ||
Theorem | addcos 15991 | Sum of cosines. (Contributed by Paul Chapman, 12-Oct-2007.) |
β’ ((π΄ β β β§ π΅ β β) β ((cosβπ΄) + (cosβπ΅)) = (2 Β· ((cosβ((π΄ + π΅) / 2)) Β· (cosβ((π΄ β π΅) / 2))))) | ||
Theorem | subcos 15992 | Difference of cosines. (Contributed by Paul Chapman, 12-Oct-2007.) (Revised by Mario Carneiro, 10-May-2014.) |
β’ ((π΄ β β β§ π΅ β β) β ((cosβπ΅) β (cosβπ΄)) = (2 Β· ((sinβ((π΄ + π΅) / 2)) Β· (sinβ((π΄ β π΅) / 2))))) | ||
Theorem | sincossq 15993 | Sine squared plus cosine squared is 1. Equation 17 of [Gleason] p. 311. Note that this holds for non-real arguments, even though individually each term is unbounded. (Contributed by NM, 15-Jan-2006.) |
β’ (π΄ β β β (((sinβπ΄)β2) + ((cosβπ΄)β2)) = 1) | ||
Theorem | sin2t 15994 | Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.) |
β’ (π΄ β β β (sinβ(2 Β· π΄)) = (2 Β· ((sinβπ΄) Β· (cosβπ΄)))) | ||
Theorem | cos2t 15995 | Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.) |
β’ (π΄ β β β (cosβ(2 Β· π΄)) = ((2 Β· ((cosβπ΄)β2)) β 1)) | ||
Theorem | cos2tsin 15996 | Double-angle formula for cosine in terms of sine. (Contributed by NM, 12-Sep-2008.) |
β’ (π΄ β β β (cosβ(2 Β· π΄)) = (1 β (2 Β· ((sinβπ΄)β2)))) | ||
Theorem | sinbnd 15997 | The sine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.) |
β’ (π΄ β β β (-1 β€ (sinβπ΄) β§ (sinβπ΄) β€ 1)) | ||
Theorem | cosbnd 15998 | The cosine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.) |
β’ (π΄ β β β (-1 β€ (cosβπ΄) β§ (cosβπ΄) β€ 1)) | ||
Theorem | sinbnd2 15999 | The sine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.) |
β’ (π΄ β β β (sinβπ΄) β (-1[,]1)) | ||
Theorem | cosbnd2 16000 | The cosine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.) |
β’ (π΄ β β β (cosβπ΄) β (-1[,]1)) |
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