FUNDAMENTAL THEOREM OF ARITHEMATIC
FUNDAMENTAL THEOREM OF ARITHEMATIC Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique apart from the order in which the prime factors occur.
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EUCLID'S DIVISION LEMMA AND ALGORITHM
EUCLID'S DIVISION LEMMA AND ALGORITHM A lemma is a proven statement used for proving another statement. An algorithm is a series of well defined steps which gives a procedure for solving a type of problem. EUCLID'S DIVISION LEMMA - Given positive integers a and b there exist unique integers q and r satisfying a=bq+r , 0≤r<b EUCLID'S DIVISION ALGORITM is a technique to compute HIGHEST COMMON FACTOR (HCF) of two positive integers. Let us take an example of euclid algorithm: Take two numbers 455 and 42 455 > 42 so, take the greater no. and apply euclid lemma 455=42 × 10 + 35 here we have Q=42,divisor=10 and remainder=35 Do these steps till the remainder become O 42= 35 × 1 +7 35=7 × 5 + 0 So, here the remainder becomes zero and algorithm works here cleary. hence, the factor of 455 and 42 is 7 class 10 euclid division
TRIGONOMETRY IDENTITIES
TRIGONOMETRY IDENTITIES Calculation of trigonometric ratios for specific angles establish some identities are called Trigonometry identities. Or can be say that an equation involving trigonometric ratios of an angle is called a trigonometry identity . SIN A = 1/COSEC A COS A = 1/SEC A TAN A = 1/COT A OR SIN A / COS A COT A = 1/TAN A OR COS A / SIN A SEC A = 1/COS A COSEC A = 1/SIN A COS 2 A+ SIN 2 A = 1 OR SIN ²A + COS²A =1 SEC 2 A - TAN 2 A = 1 OR 1+ TAN 2 A = SEC 2 A 1+COT 2 A= COSEC 2 AOR COSEC 2 A-COT 2 A=1 ADDITION AND SUBTRACTION Ø SIN (A+B) = SINACOSB +COSASINB Ø SIN (A-B) = SINACOSB – COSASINB Ø COS (A+B) = COSACOSB – SINASINB Ø COS (A-B) = COSACOSB + SINASINB Ø TAN (A+B) =TANA + TANB/1-TANATANB Ø TAN (A-B) = TANA-TANB/1 +TANATANB Ø COT (A-B) = 1-COTACOTB/COTA+COTB Ø COT (A-B) = 1+COTACOTB/COTA-COTB C & D FORMULA Ø SINC + SIND = 2SIN(C+D/2).COS(C-D/2) Ø SINC – SIND =2COS(C +D/2).SIN(