Using the identities:
We seek to prove that:
# sin x cos x(tan x + cot x) -= 1 #
Consider the LHS:
# LHS -= sin x cos x(tan x + cot x) #
# \ \ \ \ \ \ \ \ = sin x cos x(sinx/cosx + cosx/sinx) #
# \ \ \ \ \ \ \ \ = sin x cos x((sinxsinx + cosxcosx)/(sinxcosx)) #
# \ \ \ \ \ \ \ \ = sin x cos x((sin^2x + cos^2x)/(sinxcosx)) #
# \ \ \ \ \ \ \ \ = sin^2x + cos^2x #
# \ \ \ \ \ \ \ \ -= 1 \ \ \ #QED
The trp operon specifically, is repressible.
First, we must understand that there are a lot of operons out there in molecular biology that can do a variety of things.
All operons are either under positive or negative control. Positively controlled operons are ones where gene expression is only stimulated by the presence of a regulatory protein. Negatively controlled operons are ones where gene expression is turned off in the presence of a repressor - this is either repressible or inducible.
The trp operon you are talking about is a repressible system. It has multiple domains (structures that are specifically activated under certain conditions on a molecular scale).
So yes, the trp operon regulates the production of tryptophan. Let's give two situations where it is most obvious.
When levels of tryptophan are very high in the cell, levels of tryptophan RNA are also obviously very high. Therefore, immediately after translation, the mRNA will move quickly through the ribosome complex on domain 1 and the short peptide structure in tryptophan is translated very quickly.As an effect of quick translation, domain 2 in the trp operon becomes chemically associated with the ribosome complex, effectively blocking it, while domain 3 binds to domain 4, stopping translation as a loop feedback occurs. This is known as an attenuation of transcription, and only about 10% of regular mRNA is created.
On the other hand, when cellular levels of tryptophan are low, translation of the short peptide is translated slowly in domain 1. As the mRNA moves slowly, the domain 2 binds to domain 3 and because the ribosome doesn't bind to domain 2, transcription occurs normally and biosynthesis of tryptophan occurs normally.
Please see work and diagram, below.
Circle A has a center at (2, 3) and an area of 8π sq units.
Circle B has a center at (11, 7) and an area of 54π sq units.
Do the circles overlap?
Name the center of circle A: A = (2, 3).
Name the center of circle B: B = (11, 7).
1) Find-out how far apart the centers of the circles are.
2) Determine how large the radii of the two circles are.
3) If the centers are closer together than the sum of their radii, they will overlap. Otherwise, the circles may either touch (be tangent) or not be close enough either to overlap or touch.
1) Distance center-to-center:
2) Find the radii:
The sum of the two radii:
Since the distance between the centers of the circles is less than the sum of their radii, the circles must overlap.
How much overlap is there?
Diagram to help visualize the result:
Adding  and  we get
We know that,
Adding (1) and (2),
Consider the diagram:
#"Since there are 3 transformations to be performed label"#
#A(7,3)" and "B(6,5)#
#"under a rotation about the origin of "(3pi)/2#
#• " a point "(x,y)to(y,-x)#
#"under a vertical translation "((0),(3))#
#• " a point "(x,y)to(x,y+3)#
#"under a reflection in the x-axis"#
#• " a point "(x,y)to(x,-y)#
#"After all 3 transformations"#
#(7,3)to(3,4)" and "(6,5)to(5,3)#
Free verse poetry does not have rhyme or rhythm (but it's still art!).
Example (Fog by Carl Sandburg):
The fog comes
on little cat feet.
It sits looking
over harbor and city
on silent haunches
and then moves on.
Now, in the interval
The general solution is
The two can be combined in a single expression