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0

## Verify the identity sin x cos x(tan x + cot x) = 1 ?

Hi
Featured 6 hours ago · Trigonometry

Verified below

#### Explanation:

Using the identities:
$\tan x = \sin \frac{x}{\cos} x$

$\cot x = \cos \frac{x}{\sin} x$

${\sin}^{2} x + {\cos}^{2} x = 1$

Start:
$\sin x \cos x \left(\tan x + \cot x\right) = 1$

$\sin x \cos x \tan x + \sin x \cos x \cot x = 1$

$\sin x \cancel{\cos x} \cdot \sin \frac{x}{\cancel{\cos x}} + \cancel{\sin x} \cos x \cdot \cos \frac{x}{\cancel{\sin x}} = 1$

${\sin}^{2} x + {\cos}^{2} x = 1$

$1 = 1$

1

## Verify the identity sin x cos x(tan x + cot x) = 1 ?

Steve M
Featured 6 hours ago · Trigonometry

We seek to prove that:

$\sin x \cos x \left(\tan x + \cot x\right) \equiv 1$

Consider the LHS:

$L H S \equiv \sin x \cos x \left(\tan x + \cot x\right)$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sin x \cos x \left(\sin \frac{x}{\cos} x + \cos \frac{x}{\sin} x\right)$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sin x \cos x \left(\frac{\sin x \sin x + \cos x \cos x}{\sin x \cos x}\right)$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sin x \cos x \left(\frac{{\sin}^{2} x + {\cos}^{2} x}{\sin x \cos x}\right)$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = {\sin}^{2} x + {\cos}^{2} x$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \equiv 1 \setminus \setminus \setminus$ QED

2

## How does trp operon regulate tryptophan levels?

Jace Lyrafi
Featured 20 hours ago · Biology

The trp operon specifically, is repressible.

#### Explanation:

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.

1

## Circle A has a center at (2 ,3 ) and an area of 8 pi. Circle B has a center at (11 ,7 ) and an area of 54 pi. Do the circles overlap?

Nimo N.
Featured 13 hours ago · Geometry

Please see work and diagram, below.

#### Explanation:

Question:
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?

Start:
Name the center of circle A: A = (2, 3).
Name the center of circle B: B = (11, 7).

Plan:
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.

The work:
1) Distance center-to-center:

 color(blue)( abs(AB) = sqrt( (x_2 - x_1)^2 + (y_2 - y_1)^2 )
$\left\mid A B \right\mid = \sqrt{{\left(\left(11\right) - \left(2\right)\right)}^{2} + {\left(\left(7\right) - \left(3\right)\right)}^{2}}$
$\left\mid A B \right\mid = \sqrt{81 + 16}$
 color(brown)( abs(AB) = sqrt( 97 ) ~~ 9.84886 \ units

 color(blue)( "Area of circle" = pi r^2
Circle A:
 Area \ of A: \ 8 cancel(π) = cancel(π) r_a^2
$\sqrt{8} = {r}_{a}$
 color(brown)( 2.82843 \ units~~ r_a

Circle B:
 Area \ of B: \ 54 cancel(π) = cancel(π) r_b^2
$\sqrt{54} = {r}_{b}$
 color(brown)( 7.34847 \ units ~~ r_b

The sum of the two radii:
${r}_{b} + {r}_{b} \approx 2.82843 + 7.34847$
 color(brown)( r_b + r_b ~~ 10.17690 \ units

3) Conclusion:
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?
 color(blue)( "the sum of the two radii "-" the distance between the centers"
 color(brown)( 10.17690 - 9.84886 ~~ 0.32804 \ units

Diagram to help visualize the result:

1

## If tan A + sec A = 4, what is CosA?

dk_ch
Featured 6 hours ago · Trigonometry

Given $\tan A + \sec A = 4$

$\left(\sec A + \tan A\right) = 4. \ldots . \left[1\right]$

So

$\left(\sec A + \tan A\right) \left(\sec A - \tan A\right) = 4 \left(\sec A - \tan A\right)$

$\implies \left({\sec}^{2} A - {\tan}^{2} A\right) = 4 \left(\sec A - \tan A\right)$

$\implies 1 = 4 \left(\sec A - \tan A\right)$

$\implies \sec A - \tan A = \frac{1}{4.} \ldots \ldots \left[2\right]$

Adding [1] and [2] we get

$2 \sec A = 4 + \frac{1}{4}$

$\implies \sec A = \frac{17}{8}$

$\implies \cos A = \frac{8}{17}$

1

## If tan A + sec A = 4, what is CosA?

maganbhai P.
Featured 6 hours ago · Trigonometry

$\cos A = \frac{8}{17}$

#### Explanation:

Here,

$\sec A + \tan A = 4. \ldots . \to \left(1\right)$

We know that,

color(red)(sec^2A-tan^2A=1

$\left(\sec A + \tan A\right) \left(\sec A - \tan A\right) = 1$

$\implies \left(4\right) \left(\sec A - \tan A\right) = 1. \ldots . \to$From(1)

$\therefore \sec A - \tan A = \frac{1}{4.} \ldots . \to \left(2\right)$

$\sec A + \cancel{\tan} A = 4$

(secA-canceltanA=1/4)/

$2 \sec A = 4 + \frac{1}{4} = \frac{17}{4}$

$\implies \sec A = \frac{17}{8}$

$\implies \cos A = \frac{8}{17}$

2

## A 24 foot tree casts a shadow that is 9 feet long . At the same time a nearby building cats a shadow 45 feet long . how tall is the building?

Gió
Featured 6 hours ago · Trigonometry

I got $120 \text{ft}$

#### Explanation:

Consider the diagram:

The angles $\alpha$ are the same so we can write for the two triangles (tree and building):
$\frac{h}{l} = \frac{H}{L}$
in numbers:
$\frac{24}{9} = \frac{H}{45}$
so that:
$H = \frac{24}{9} \cdot 45 = 120 \text{ft}$

1

## A line segment has endpoints at (7 ,3 ) and (6 ,5). If the line segment is rotated about the origin by (3pi )/2 , translated vertically by 3 , and reflected about the x-axis, what will the line segment's new endpoints be?

Jim G.
Featured 13 hours ago · Geometry

$\left(3 , 4\right) \text{ and } \left(5 , 3\right)$

#### Explanation:

$\text{Since there are 3 transformations to be performed label}$
$\text{the endpoints}$

$A \left(7 , 3\right) \text{ and } B \left(6 , 5\right)$

$\textcolor{b l u e}{\text{First transformation}}$

$\text{under a rotation about the origin of } \frac{3 \pi}{2}$

• " a point "(x,y)to(y,-x)

$\Rightarrow A \left(7 , 3\right) \to A ' \left(3 , - 7\right)$

$\Rightarrow B \left(6 , 5\right) \to B ' \left(5 , - 6\right)$

$\textcolor{b l u e}{\text{Second transformation}}$

$\text{under a vertical translation } \left(\begin{matrix}0 \\ 3\end{matrix}\right)$

• " a point "(x,y)to(x,y+3)

$\Rightarrow A ' \left(3 , - 7\right) \to A ' ' \left(3 , - 4\right)$

$\Rightarrow B ' \left(5 , - 6\right) \to B ' ' \left(5 , - 3\right)$

$\textcolor{b l u e}{\text{Third transformation}}$

$\text{under a reflection in the x-axis}$

• " a point "(x,y)to(x,-y)

$\Rightarrow A ' ' \left(3 , - 4\right) \to A ' ' ' \left(3 , 4\right)$

rArrB'')5,-3)toB'''(5,3)

$\text{After all 3 transformations}$

$\left(7 , 3\right) \to \left(3 , 4\right) \text{ and } \left(6 , 5\right) \to \left(5 , 3\right)$

1

## Do poems have to have rhythm or rhyme?

lechugamaestro19
Featured 9 hours ago · English Grammar

No.

#### Explanation:

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.

1

## How do you solve cos^-1(tan x) = pi?

Ananda Dasgupta
Featured 6 hours ago · Trigonometry

$x = \left(4 n + 3\right) \frac{\pi}{4} , q \quad n \in \mathbb{Z}$

#### Explanation:

${\cos}^{-} 1 \left(\tan x\right) = \pi \implies \tan x = \cos \pi = - 1$

So $x = {\tan}^{-} 1 \left(- 1\right)$

Now, in the interval $\left[0 , 2 \pi\right)$, there are two angles that satisfy $\tan x = - 1$ - these are $x = \frac{3 \pi}{4}$ and $\frac{7 \pi}{4}$.

The general solution is

$\frac{3 \pi}{4} + 2 n \pi$ or $\frac{7 \pi}{4} + 2 n \pi$, $q \quad n \in \mathbb{Z}$

The two can be combined in a single expression

$x = \left(4 n + 3\right) \frac{\pi}{4} , q \quad n \in \mathbb{Z}$