Solutions To Trigonometric Equations (Part One)

 Solve the following equations for values of q in the range:

0  <  q   < 2p.

1) 2 cos  q  - 1  = 0

Solution:

Transpose to get angle on one side:

2 cos q  = 1

cos q  = 1/2

q  =  p/ 3 (From table)

And:   q  =  5p/ 3   for next quadrant, i.e.

2) Ö3 tan² q  + 2tan q  - Ö3   =  0

Solution:

Rewrite as quadratic eqn. in x:

Ö3 x²   + 2 x   - Ö3   =  0

Solve using the quadratic formula:

 a =Ö3, b = 2, c = -Ö3  we have:

 b ²  - 4ac    =   4 - 4(-3)  =  16

Substituting values for the full formula:

x =  - 2 +  Ö16/  2Ö3  = - 2 +4   /  2Ö3  = -6/  2Ö3 ,  2/ 2Ö3 ,

= -3/ Ö3 ,  1/Ö3 ,

x = - Ö3,  x = Ö3/3 

Substituting back angle functions:

  tan q = - Ö3,  tan q = Ö3/3

For: tan q = Ö3/3, q = p/ 6,  7p/ 6

For: tan q = -Ö3,  q = - p/ 3 (not in range), 2p/ 3,  5p/ 3

So angle solns over range 0  <  q   < 2p:

p/ 6,  7p/ 6, 2p/ 3,  5p/ 3

3) 2Ö3 csc ² q - 7 csc  q  +  2Ö3   =  0

Solution:

Rewrite as quadratic eqn. in x:

2Ö3 x²   - 7 x   + 2Ö3   =  0

Solve for x using the quadratic formula:

a =2Ö3, b = - 7, c = 2Ö3  we have:

 b ²  - 4ac    =   49 - 4(12)  =  1

Substituting values for the full formula:

x = 7  +  1/4Ö3  =  8/4Ö3  ,   6 /4Ö3 

=  2/Ö3,   3/  2/Ö3  =   2Ö3/3,   Ö3/2

Substituting back angle functions:

For: csc q = 2Ö3/3, q = p/ 3  (from table) 

And also:

For: csc q = Ö3/2 = 0.866, q does not exist 

Rem: csc q   = 1/ sin q   so sin q =  1/ 0.866 =1.1547

Which is impossible,  i.e. NO sin q >1

4) sec ² q - 3 sec q + 2 = 0

Solution: Factor as a regular quadratic eqn.

(sec q - 2) (sec q - 1) = 0

sec q - 2 = 0

sec q = 2 

q =   p/ 3  (From table)

Also include:  q =   5p/ 3 ,  7p/ 3

For 2nd factor:

sec q - 1 =  0

sec q =1  

q =   0 ⁰  

5) sec 3 q + 2 = 0

Solution:

Transpose sec to one side:

sec 3 q = - 2 

  Convert to cosine function and take reciprocal of both sides:

(Since: sec q = 1/ cos q)

Then:     cos 3 q =   - ½

  This function is -ve in quadrants 2, 3 so:

              3q =    2p/ 3,   4p/ 3,                             

From here other q solns. can also be found as per the method shown in Part 1, ex. 4, i.e.

The table below was used to obtain the angles from the trig functions:

Trump's Elevation Of Stephen Miller Shows He's More Invested In Combat Than Cooperation - Buckle Up For More Chaos

 

                            "America's Hitler' - needs checking by congress.

                 Stephen Miller, (right) and kindred spirit Joseph Goebbels

“Trump’s elevation of Stephen Miller because he supposedly gets things done is the reshuffle version of a middle finger. What does Trump want his new White House chief of staff to do? Miller’s specialism is presidential rule without regard for Congress or the courts. He is the unmasked face of immigrant deportations, champion of history’s “iron laws” of might makes right and of an unbound US presidency. Anyone who was hoping Trump would play the role of a lame duck instead of an angry duck has been put on notice.” Edward Luce, ‘Trump Takes The Alama Route’,  Financial Times

"Tonight, Trump sent an unmistakable signal that he is prepared to do battle with the Democratic congress by ousting Susie Wiles and replacing her with Stephen Miller .  Miller is the combative architect of the ICE immigration raids, invading blue cites and mass deportations, and directing the military to bomb boats in the Caribbean."  - Jonathan Karl, from ABC News Friday night.

Those of us with more than air between the ears always knew that when Dotard said he was willing to work with the Dem congress  (after the shellacking he took Nov. 3rd) it was bollocks. Similar to when he said he would 'change his tone' after taking a would-be assassin's bullet in the ear back in 2024. Did he do so? No way in hell, see e.g.

Donald Trump

Nothing Will Change Trump — Not Even an Assassination Attempt

The reason is that this walking orange offal is a degenerate, decomposing, depraved traitor who will always believe himself above all human laws. Especially after the Roberts court  - as Jamie Raskin noted on All In Friday - "granted him immunity for any crimes committed in office."  So, of course that includes picking other criminals and traitors to do his bidding.  Reinforced when Miller blurted Friday night on FOX:

"The mission now is more clear than ever. It's to fight with all our soul to defend our president and defend his magnificent republic."

Of course, the last words should read: "magnificent criminal regime". What these MAGA misfits have created is hardly a republic. Running a de facto dictatorship - invading American cities, using masked thugs to kill citizens - is definitely alien to any republic, especially when the word is uttered by a scumball like Miller. As new Dem House Majority leader Hakeem Jeffries put it:

"Stephen Miller is a dangerous xenophobe and a reckless extremist who has American blood on his hands. He should be removed from acting in violation of laws like a malignant cancer."

The shitbag demented Trump actually confirmed the worst suspicions of pundits that he was giving a "middle finger to the electorate".  This, when he appointed Miller as his WH consigliere then raged that he wanted Dems "imprisoned" for wanting to hold him accountable.  So this lunatic traitor still thinks he's going to run around unchecked.  In the meantime, Dems have to be aware of what he has planned in the nearly two months before they are seated on Jan. 3rd. According to The Wall Street Journal ('Trump Faces New Reality', Nov. 6, p.A1):  

"Aides including Stephen Miller and Office of Management  and Budget Director Russell Vought, have already been planning executive orders that Trump can issue quickly in key areas including immigration. trade, housing and reducing the federal workforce.  Trump also plans to continue exerting enormous influence over the Justice Department, which he uses to target political opponents."

Clearly, this deranged, malignant narcissist hasn't learned a damned thing from the Nov. 3rd election, and even Thom Tillis (R, NC) asserted after the Miller elevation that "this was the wrong signal to send."  Yuh think?

But see, the upstart Trumper criminal renegades simply don't care. With the DOJ working for them, they totally believe they are beyond any challenge or accountability.  Well, we will see as this unfolds and whether these lunatics test the limits of their power - say by ignoring subpoenas-  against a determined, massive election blue wave congress soon coming into D.C.

But no one should get too overtaken by fantasies and believe the political landscape will now suddenly be healed and bipartisanship renewed. Trump's elevation of his architect of hate Miller to the status of literal 'co-president' (in terms of his power) pretty much ensures we're in for more partisan nastiness, dysfunction and division until (hopefully) 2028 finally rids us of Trump's toxic presence. And the GOP stabilizes itself as a political party as opposed to one deranged man's personality cult.

In the meantime, how and when the Dem congress wields its power - and whether it can enforce that power-  will determine if we still have any kind of democracy, say by the Spring of 2027. The plea to "prioritize people over investigations" is noble to be sure, but somehow missed the other polling message sent by voters. In the 2026 SSRS Voter Poll, conducted on Election Day, 90% of voters said that corruption was a crucial or major problem, at least as important as inflation, immigration enforcement, and the war in Iran. See e.g.

Midterm voters tell politicians: Tackle corruption - CSMonitor.com

So clearly, we can't allow Trump's lawless renegades to break down whatever remaining barriers there are to the nation's stability.  And that includes Trump AG Todd Blanche himself threatening to "prosecute" lawmakers who try to hold these maggots accountable by pursuing no holds barred investigations.

The rest of the world, indeed, remains distrustful that even one momentous election result will release it from the maw of Trump's derangement and depravity. I.e. 'World Struggling To Trust U.S. After Midterm Results', Associated Press, noting: 

"Interviews by the Associated Press around the world revealed a sense that 'one election alone does not restore anything. We have to see what they actually do in terms of gaining at least a little trust back.'  That 77 million Americans voted to elect Trump to a second term - along with Trump's combative tariff war, enabling Israel's conduct in Gaza, cruel immigration policies and starting a war in Iran - had already shaken trust around the world."

In other words, all of these need to be rectified, and that's only possible by checking Trump's power - which may also require targeted investigations.

See Also:

Trump unravels with wild demand for Democrats’ imprisonment

And:

For Stephen Miller, Election Defeat Is No Reason to Back Down on Immigration Crackdown - The New York Times

Excerpt:

Republicans called for President Trump to scale back the policies many blamed for their midterm blowout. Mr. Trump ruminated about deals with Democrats.

Stephen Miller was unfazed.

A day after the elections, as votes were still being counted and the Democrats’ victory margin seemed to increase by the hour, the longtime Trump aide and MAGA stalwart convened one of his regular calls with national security officials to get an update on immigration policy.

There was no talk of the results of the midterms or suggestion of scaling back the hard-line immigration approach that contributed to Republicans’ steep losses, according to people with knowledge of the call, who spoke on the condition of anonymity to describe a private discussion. Instead, Mr. Miller continued to push the officials to carry out an agenda that has in the past year included ramping up deportations, driving down border crossings and removing immigrants to faraway nations they were not from.

In other words, the architect of some of the administration’s most divisive policies demonstrated that he would aggressively pursue more of the agenda that many voters rejected on Tuesday.

Now, after being named Friday as White House chief of staff, Mr. Miller is poised to help oversee what is likely to be a combative final two years of Mr. Trump’s term. And as Democrats vow to investigate the very policies that Mr. Miller helped devise, Mr. Miller, the ultimate Trump loyalist, is in a prime position to continue his longtime pursuit of limiting checks on executive power.

Democrats on Friday indicated that, as they prepare to scrutinize the White House, Mr. Miller himself is squarely in their sights. Representative Robert Garcia, who is set to lead the House Oversight Committee, called Mr. Miller the “architect of the unconstitutional ICE raids that have terrorized our communities.”

And:

Stephen Miller’s plot for sickening crackdown on free speech is leaked

And:

Now We Act: With Rep. Ro Khanna on America’s Future

And:

Stephen Miller

Likely ‘To Go Very Badly’: Critics Gobsmacked As Miller Becomes Chief of Staff

by Sarah K. Burris | November 7, 2026 - 6:17am | permalink

— from Alternet

President Donald Trump announced on Truth Social that he would be turning a new page with a new chief of staff in the White House for the new Democratic Congress. Susie Wiles is out and Stephen Miller is in.

"I am pleased to announce that Mr. Stephen Miller, the current Deputy Chief of Staff of the Trump Administration, and one of my most trusted advisors, will become the Chief of Staff starting Monday, November 9, 2026. Susie Wiles, who has done an outstanding job, will be moving over to the Trump Political Operation. I want to thank her for her service as Chief of Staff — She has done a great job!" Trump wrote online before signing his name.

If Democrats thought there was a chance of bipartisanship and working together with the White House, Miller will make that very difficult.

» article continues...

And:

In Elevating Stephen Miller, Trump Doubles Down on Hard-Line Agenda - The New York Times

Excerpt:

He said he would work with Democrats, claiming as he has over the years that he could forge a deal with them on immigration, perhaps the most polarizing domestic issue of his second term.

But near the week’s end, instead of ceding to the clear evidence of voter displeasure, Mr. Trump pivoted to a familiar move: more MAGA.

On Friday, the president ousted his onetime close adviser, Susie Wiles, from the role of White House chief of staff, replacing her with Stephen Miller, his deputy chief of staff. More than any other top adviser, Mr. Miller encouraged an aggressive flexing of executive power when Mr. Trump returned to office.

Mr. Miller has been the driver of the president’s most muscular actions, including the use of increasingly hostile tactics to capture immigrants across the country and the bombing of small boats in the Caribbean Sea and the eastern Pacific Ocean that he and Mr. Trump declared constituted a foreign invasion.

The president’s choice sent a clear signal about the posture he intends to strike in his final two years in office and how little he intends to alter his approach in response to the most decisive political rout in half a century. He also put on center stage a figure who had already been a galvanizing target for Democrats intent on using their new powers in Congress to check the administration.

 

And:

Stephen Miller

Trump Makes His Chief Bigot His Chief of Staff

by Robert Reich | November 7, 2026 - 6:11am | permalink

— from Robert Reich's Substack

Friends,

Anyone wondering how Trump will respond to the biggest repudiation a modern U.S. president has ever received from American voters need wonder no longer. Today, Trump announced that Stephen Miller would become his chief of staff, replacing Susie Wiles.

In a Truth Social post, Trump called Miller one of his “most trusted” advisers and said he would be starting in the new role Monday.

Miller as chief of staff is not a positive development for anyone interested in mending our government or healing our nation.

As Trump’s attention recedes into his vanity projects — the ballroom, the arch, the heliport, the golf links, the gold-leaf makeovers — Miller is likely to become de facto president, manipulating an increasingly incoherent and cognitively impaired old man.

» article continues...

And:

Republicans ‘very worried’ Stephen Miller is now the ‘face’ of the party

An In Depth Treatment Of The Landau Contour - And The Role Of the Inverse Laplace Transform. (Part 2)

Recall at the end of Part 1, I noted Landau's contribution:

What Landau did was to extend e (k,w) to the entire w- plane. In velocity space the pole can be anywhere in the upper right quadrant (since w _i >  0). To make e (k,w) an entire function one needs to follow an analytic continuation process, i.e. from the upper half plane to the lower half plane.  

Thus we need:

i) I =   ò ^¥_-¥  ^du f_o /(u  -  w/ k)                              (w >0)

ii) I =   ò ^¥_-¥  ^du f_o/(u   -  w/ k)  + 2pi (Res)      (w _i > 0)

These will feature later on when we determine the Cauchy principal value P for the quantity:   e(k, wr) 

We look next at the dielectric function to see if any damping is associated with normal modes. We find:

e  =1 - w_e ²/ k^{2   ò  du} f_o  / ( u   -  w/ k)    

Note the dielectric function is equivalent to the Poisson equation: 

ik E  =  4pr

Now consider:

E (w ) =  4pe/ k² e (k,w)   ò  dv f₁  (k, v, t= 0)/  v _x-  (w/ k)

analytically continued to the entire complex   w -plane.  Then the L-contour will be deformed as shown in the graphic below:

No half plane attempt is made to close the contour, i.e. with a semi-circle in the lower  w-plane since it isn't clear that E (w) falls fast enough at large negative  w_i  for this to be done. The alternative method is to treat each part of the contour separately. In this case four types of contributions emerge but we focus on one in this post.

 That is, the poles of  E (w) such as the one at A.  Assume the pole comes from a zero of E (w)  instead of the factor on the right of the equation. We can denote the frequency at A by w_A  - assuming it originates from the zero at E (w).  Then the pole contributes a term to E(t) with time dependence, exp (-iw_A t).

These types of contributions are called normal modes and since their frequencies are given by  E (w)= 0 they correspond to waves found by solving the dispersion equation for electrostatic waves in unmagnetized plasma, i.e.:

1 + w_e²/ k^{2   ò  du  d} u g(u)/  u   -  w/ k = 0

The dominant normal mode contribution comes from the mode with the largest imaginary part of the frequency, e.g. A. if positive the normal mode is unstable and grows with time. If negative, the mode is damped and decays with time. Typically, one looks at the dielectric function to see if any damping is associated with the normal modes, i.e.

 e(k, w)   = 1 -  w_e²/ k^{2   ò  du  d} u g(u)/  u   - ( w/ k) 

We rewrite the dispersion relation (which we will also use at the end) as:     e (k, w) =   er   +  i er    = 0

Or:   e (k, w) =  0

We have:  w_i =  e i (k, wr)/ {¶ e r (k, w )/¶ w] w= wr   

When w_i = 0, the Landau contour appears as shown below:

  It is instructive to perform the integrations from just above the pole to just below to show the explicit form for the Cauchy principal value, P.

From just above the pole:

I ⁺ =  ò ^¥_-¥  ^du f_o /  (u  -  w/ k -  id) 

From just below:

I ^- =  ò ^¥_-¥  ^du f_o /  (u   -  w/ k -  id) +  2pi (Res)

Take the average:   

½ [ I ⁺ +  I ^- ]  Þ 

lim d _{® 0}   [ò ^¥_-¥  ^du f_o / (u  -  w/ k -  id) + ò ^¥_-¥  ^du f_o /  (u  -  w/ k + id)] +pi (Res)

Where the (purple) bracketed quantity is P the Cauchy Principal Value.   Use of the Cauchy principal value allows us to obtain:

e(k, wr)    = 

1 -  w_e²/ k^2    P ò ^¥_-¥ ^{du  (du} u g(u))/  (u   -  w/ k) 

-  pi  w_e²/ k^2    du u g(u)

 The 2nd term on the right (shown directly above) comes from integrating around the 'mini' semicircle for the previous Landau contour.  This yields one half of 2 pi  times the residue.

In normal mode (steady state) behavior, we only need Landau's prescription - so there's no need to go back to the Laplace transforms- which actually contain more information than needed.

From the previous equation:

 er(k, wr) = 0

and:

e(k, w)   = 1 -  w_e²/ k²   ò  ^du  d u g(u)/  u   - ( w/ k) 

We can determine the real part of the frequency  wr  from:

e(k, wr)    = 

1 -  w_e²/ k^2    P ò ^¥_-¥ ^{du  (du} u g(u))/  u   - ( w/ k)  

-  pi w_e²/ k^2    du u g(u)|u= /k 

 For the purposes of this integration we take: 

^du u g(u) = 0  at   u   =  w_r/ k

Then integrate by parts to get:

e(k, wr)    =   1 -  w_e²/w_r² - {3k²   v_0e² w_e² /w_r⁴ }  =  0

Where:  v_0e  =  v_thermal   - which for the Langmuir wave dispersion relation:

w²   = w_e² + 3k²  v_oe² 

yields:  w_r²   = w_e² + 3k²  v_oe² 

with the assumption:

 k²   v_oe²     <<  w_r²

Taking a Taylor expansion about  w_r yields:  

e (k, wr) + (w  -  wr )¶ er/ ¶ w] wr + i e i + i (w  -  wr ) ¶ e i/ ¶ w

Solving the non-zero part:

Þ

i er (k, wr)  +  i wi ¶wi  ¶ w] wr + i e  = 0

er= 0     Þ   wr  =  wi   =  -  ei  /¶ er/ ¶ w] wr   

=    pi  w_e²/ k^2   (df_o^/du) 1/(¶ er/ ¶ w) wr   

w_r²  = w_e² + 3k²  v_oe²

Which returns us to the Langmuir dispersion relation obtained earlier via integration by parts. 

Note that:  

(¶er/ ¶ w) ] wr 

 »

  2 w_e²/w_r3  

 »  

2 /w_e

Then by:    w_i =  e i (k, wr)/ {¶ e r (k, w )/¶ w] w= wr  

And: e(k, wr)   = 1 -  w_e²/ k^{2   P} ò ^¥_-¥ ^{du  (du} u g(u))/ (u  -  w/ k) 

-  pi  w_e²/ k^2    du u g(u)]u= w/k

Þ

w_i   = p w_e³/ 2k^2   (du u g(u)]u= w/k

And the total normal mode frequency is given by:

 w   = w_e (1 + 3/2  k²   l_e²) + i p w_e³/ 2k^2  (du u g(u)]u= w/k

Assume now that the phase velocity is large compared to the thermal velocity, i.e.      vf ² >>  v_oe²

Then:  er  = 1 -  w_e²/w²

Þ     (¶er/ ¶ w)   =  2 w_e²/w_r3    +  6k²   v_0e²/w_r³

  k²   v_0e²/w_r³     =  k²   l_D²/w_e

 w_i   =  -  ei  /¶ er/ ¶ w] wr   =  p w_e²/ k^2   (df_o^/du) (1/ 2 w_e²/w_r3 )

=  {p/2}  w_e³/k^2   df_o/du]u= v f

Note from the preceding that the damping rate depends significantly on the slope of the distribution function, i.e.

If then the phase velocity ( vf) or f_o (u) shows negative slope you get damping.  Hence, if there are more slower than faster plasma particles one gets a net energy transfer from wave -> particle.

In Part 3 I look at the transient (energy) effects on normal modes for the three-way deformed Landau contour and also obtaining the plasma dispersion function.