Proofs and identities Flashcards
(15 cards)
1
Q
Sin^2 X + Cos^2 X
A
=1
2
Q
Sec X
A
=1/Cos X
3
Q
Cosec X
A
=1/Sin X
4
Q
Cot X
A
=1/Tan X
=Cos X/Sin X
5
Q
Tan^2 X + 1
A
=Sec^2 X
6
Q
1 + Cot^2 X
A
=Cosec^2 X
7
Q
Tan X
A
=Sin X/Cos X
8
Q
Sin 2X
A
=2Sin X Cos X
9
Q
Cos 2X
A
=Cos^2 X - Sin^2 X
=2Cos^2 X - 1
=1 - 2Sin^2 X
10
Q
Tan 2X
A
=2Tan X/1 - Tan^2 X
11
Q
Root 2 is irrational contradiction
A
Root 2 = a/b 2 = a^2/b^2 a^2 = 2b^2 Since 2b^2 is even let a=2k (2k)^2 = 2b^2 4k^2 = 2b^2 2k^2 = b^2 Since 2k^2 is even, b^2 is also even Since b^2 is even, so is b Two even numbers cannot be relatively prime So root 2 cannot be expressed as a rational fraction Therefore root 2 is irrational
12
Q
Infinite prime numbers contradiction
A
Assume there are finite prime numbers
Thus P1, P2, P3…Pn
Let N = P1 * P2 * P3 *…Pn + 1
Therefore N is not divisible by any of the prime numbers
Because you’re always left with a remainder of 1
Thus N must be divisible by another prime not on the list
This contradicts that there is a finite number
Therefore there must be infinite primes
13
Q
Sin X - small angle approximation
A
X
14
Q
Cos X - small angle approximation
A
1- X^2 /2
15
Q
Tan X - small angle approximation
A
X
