LinneΜuniversitetet Institutionen foΜr datavetenskap, fysik och matematik Formelsamling till kursen GrundlaΜggande matematik Trigonometriska formler sin(−π₯) = − sin π₯ sin(π − π₯) = sin π₯ sin(π/2 − π₯) = cos π₯ sin(π₯ + π¦) = sin π₯ cos π¦ + cos π₯ sin π¦ sin π₯ cos π₯ cos(−π₯) = cos π₯ cos(π − π₯) = − cos π₯ cos(π/2 − π₯) = sin π₯ cos(π₯ + π¦) = cos π₯ cos π¦ − sin π₯ sin π¦ sin 2π₯ = 2 sin π₯ cos π₯ 1 − cos 2π₯ sin2 π₯ = 2 cos 2π₯ = 2 cos2 π₯ − 1 = 1 − 2 sin2 π₯ = cos2 π₯ − sin2 π₯ 1 + cos 2π₯ cos2 π₯ = 2 sin2 π₯ + cos2 π₯ = 1 tan π₯ = π₯ 0 cos π₯ 1 sin π₯ 0 π 6 √ 3 2 1 2 π 4 π 3 1 √ 2 1 √ 2 1 √2 3 2 π 2 π 0 −1 1 0 Potens– och logaritmlagar Om π, π > 0 saΜ gaΜller: π0 = 1 ππ₯+π¦ = ππ₯ ⋅ ππ¦ (ππ₯ )π¦ = ππ₯π¦ (π ⋅ π)π₯ = ππ₯ ⋅ ππ₯ ππ₯ ππ₯−π¦ = π¦ ( π )π₯ πππ₯ = π₯ π π π > 0, π β= 1 saΜ gaΜller: π π log 1 = 0 log(π₯ ⋅ π¦) =π log π₯ +π log π¦ π₯ π π log( ) =π log π₯ −π log π¦ log(π₯π¦ ) = π¦ π log π₯ π¦ Fakulteter och binomialkoeο¬cienter FoΜr varje heltal π = 1, 2, 3, . . . aΜr π! = π(π − 1) ⋅ ⋅ ⋅ 3 ⋅ 2 ⋅ 1. FoΜr π = 0 aΜr 0! = 1. ( ) π! FoΜr alla heltal π, π med 0 ≤ π ≤ π aΜr ππ = π!(π−π)! Om Binomialteoremet FoΜr varje naturligt tal π gaΜller ( ) ( ) ( ) ( ) π ( ) ∑ π π−π π π π 0 π π−1 1 π π−2 2 π 0 π π (π₯ + π¦) = π₯ π¦ = π₯ π¦ + π₯ π¦ + π₯ π¦ + ⋅⋅⋅ + π₯π¦ π 0 1 2 π π=0 PolaΜr form och potensform Varje komplext tal π§ = π₯ + ππ¦ kan skrivas π§ = ππππ = π(cos π + π sin π).