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Proofs
Proof means sufficient evidence to establish the truth of something. It is obtained from deductive reasoning, rather than from empirical arguments. Proof must show that a statement is true in all cases, without a single exception.
1,294 Questions
Can you Show 1 over sinx cosx - cosx over sinx equals tanx?
From the Pythagorean identity, sin2x = 1-cos2x.
LHS = 1/(sinx cosx) - cosx/sinx
LHS = 1/(sinx cosx) - (cosx/sinx)(cosx/cosx)
LHS = 1/(sinx cosx) - cos2x/(sinx cosx)
LHS = (1- cos2x)/(sinx cosx)
LHS = sin2x /(sinx cosx) [from Pythagorean identity]
LHS = sin2x /(sinx cosx)
LHS = sinx/cosx
LHS = tanx [by definition]
RHS = tanx
LHS = RHS and so the identity is proven.
Q.E.D.
What is 3095sixteen to a base ten numeral?
You can convert this to base ten by re-writing 3096 as a summation of hex powers:
3*16^3 + 0*16^2 + 9*16^1 + 5*16^0 = 12437 in base 10
G is finite group if and only if the number of it is subgroups is finite?
False. G may be a finite group without sub-groups.
What are the steps performed by Demorgan's theorem?
DeMorgan's theorem can be simply be stated simply as "Break the line, and change the sign." Where line represents complement...
Prove that the inverse of an invertible mapping is invertible?
Assume that f:S->T is invertible with inverse g:T->S, then by definition of invertible mappings f*g=i(S) and g*f=i(T), which defines f as the inverse of g. So g is invertible.
Is every abelian group is cyclic or not and why?
every abelian group is not cyclic. e.g, set of (Q,+) it is an abelian group but not cyclic.
What is the weight resistance of a gallon jug filled with air under water?
The buoyant force is equal to the weight of the displaced fluid (Archimedes Principle). A gallon jug underwater will have a buoyant force equal to the weight of the displaced water: eight pounds.
The total weight will also include the normal downward weight of the jug itself and the air it contains.
The trivial subsets of a set are those subsets which can be found without knowing the contents of the set.
The empty set has one trivial subset: the empty set.
Every nonempty set S has two distinct trivial subsets: S and the empty set.
Explanation:
This is due to the following two facts which follow from the definition of subset:
Fact 1: Every set is a subset of itself.
Fact 2: The empty set is subset of every set.
The definition of subset says that if every element of A is also a member of B then A is a subset of B. If A is the empty set then every element of A (all 0 of them) are members of B trivially. If A = B then A is a subset of B because each element of A is a member of A trivially.
What kind of rock can you see forming in puuoo crater?
The kind of rock you see forming in Puu Oo crater is extrusive igneous. This type of rock is formed by lava.
If all x are not used is f a function?
As I understand the question: yes, f(x) can be a function even if f(x) is not defined for all x. For example, f(x) = x/x is a function that is equal to 1 everywhere but at x=0, where it is undefined.
Divide by 100. ie move decimal point 2 spots to the left, so 158cm becomes 1.58m
How do you prove that the derivative of sec x is equal to sec x tan x?
Show that sec'x = d/dx (sec x) = sec x tan x.
First, take note that
sec x = 1/cos x;
d sin x = cos x dx;
d cos x = -sin x dx; and
d log u = du/u.
From the last, we have du = u d log u.
Then, letting u = sec x, we have,
d sec x = sec x d log sec x; and
d log sec x = d log ( 1 / cos x )
= -d log cos x
= d ( -cos x ) / cos x
= sin x dx / cos x
= tan x dx.
Thence, d sec x = sec x tan x dx, and
sec' x = sec x tan x,
which is what we set out to show.
There is no simple answer and no single proof. Indeed, for some values of n, the answer is still an unproven conjecture.The smallest radius, r, in which c circles can be packed is as follows:
c = 1: r = 1
c = 2: r = 2
c = 3: r = 1+2/3*sqrt(3) = 2.154...
c = 4: r = 1 + sqrt(2) = 2.414...
c = 5: r = 0.6854...
c = 6: r = 3
c = 7: r = 3
c = 8: r = 1 + 1/sin(pi/7) = 3.304...
c = 9: r = 1 + sqrt[2*(2 + sqrt(2)] = 3.613...
c = 10: r = 3.813...
All these are obvious or proven.
In the next ten, only 11, 12, 13 and 19 have been proven. The conjecture for 20 is 5.122....
How can I prove an angle bisector?
It depends on what is given.
In general, one half of the bisected angle is proven to congruent to the other half. By the Definition of an Angle Bisector, the bisected angle can be proven bisected.
---- To show that two angles are congruent:
One way to prove the two angles congruent is to show that their measures are equal. This can be done if there are numbers on the diagram. Use the Protractor Postulate or the Angle Addition Postulate to find the smaller angles' measures, if they are not directly marked. Then use the Definition of Congruent Angles to prove them congruent.
Given that the smaller angles correspond on a congruent or similar pair of figures in that plane and form an angle bisector, the Corresponding Parts of Congruent Figures Postulate or Corresponding Parts of Simlar Figures Postulate may be used.
