Solving Trigonometric Equations (Part One)

In many ways solving trigonometric equations is similar to solving ordinary algebraic equations, including techniques such as transposition, application of the distributive law and even quadratic equations.  In this two part series I look at ways of solving these equations which often have application in advanced astronomy.  The main objective will be to find angles (specifically between 0 and 2p ) which satisfy equations involving trigonometric identities.  Thence, each angle which satisfies such an equation will be called  solution of the trigonometric equation.

Basically, to solve a trigonometric equation for the angle we first solve it for a function of the angle and then find the angle by use of tables. I include the table below which can be used to obtain most angles:

Now some example problems and their solutions:  

Example (1)  

Solve:  4 tan q  = Ö3  +  tan q

Solution:  

Transpose  tan q  and combine coefficients:

4 tan q   -  tan q  =  Ö3

3 tan q  =  Ö3

 tan q  =  Ö3/  3

From the table  q  = p/ 6, and  q  = 7p/ 6

Example (2)  

Solve:  3 sec² q  - 5 sec q  - 2 =  0

Looking carefully one can see the resemblance in structure to a quadratic eqn.

I.e. replace  sec q with x and you see:

3x ²   - 5x - 2 = 0

Which can be factored to yield:

(3x + 1) (x - 2) = 0  

With x = 2 and x = -1/3  

In the same way the trigonometric eqn. can be factored:

(3 sec q + 1) (sec q - 2) = 0   

To obtain q: sec q = 2, sec q = -1/3

If sec q = 2 we find from the table  = q  = p/ 6

For the secondary quadrant value:

Example (3)  

Solve:   tan² q  - tan q  =  0

Solution: Factor left side of the eqn.

tan q (tan q  - 1)  = 0

Setting each factor = 0:

tan q = 0 ,  (tan q  - 1)  = 0

tan q = 0 ,  1

The allowable angle solutions (from the table) are:

q = 0, p,  p/ 4  and  5p/ 4  for extra quadrant

Example (4)  

Solve:   2 sin q cos q - 2 cos 2 q  - sin q + 1  = 0

Solution: 

The left side can be factored to obtain:

(sin q - 1)(2cos 2 q  - 1) = 0

 2 angles appear in this equation but each factor contains only one function of one angle.  Setting the first factor = 0:

sin q - 1 = 0 

sin q = 1

q = p/ 2

The 2nd factor contains the angle 2 q  and we want values of   q in the range: 0  <  q   < 2p.   Hence we need to find values of 2 q  in the range: 0  <  2q   < 4p.  The 2nd factor equated to 0 gives:

2 cos 2 q  - 1  = 0

Hence:   cos 2 q =   1/2

And:   2q = p/ 3,  5p/ 3, 7p/ 3,  11p/ 3, 

q = p/ 6,  5p/ 6, 7p/ 6,  11p/ 6, 

Practice Problems:

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

0  <  q   < 2p.

1) 2 cos  q  - 1  = 0

2) Ö3 tan² q  + 2 tan q  - Ö3   =  0

3) Ö3 cos  q tan q - 3 cos  q  -  tan q  +  Ö3   =  0

4) 2 sin q cos 2q - sin q  - 2 cos 2 q  + 1  = 0

5) sec 3 q + 2 = 0

(Use the table provided to obtain the angles from the trig functions).

Another Jarring Time Change: Time To Jettison Daylight Savings Once And For All

"It’s not one hour twice a year. It’s a misalignment of our biologic clocks for eight months of the year. When we talk about DST and the relationship to light we are talking about profound impacts on the biological clock, which is a structure rooted in the brain. It impacts brain functions such as sleep-wake patterns and daytime alertness,”-   

"'The 'Sunshine Protection Act' doesn't provide additional daylight. While we may associate Spring with more daylight, that's a result of the natural lengthening of days, not the changing of the clock. All the new law would do is move one hour of darkness from the evening to the morning. In Fargo, ND under year round DST, sunrise on Dec. 21 would be after 9 a.m. for example."  Lisa Lewis, LA Times

"We should be ending daylight saving time, not making it permanent. Mornings have been miserable for our family as we wake up in the darkness. I can’t imagine having to do this all winter. If we’re not going to move our clocks, then standard time makes more sense. Our clocks should read 12 p.m. as close to solar noon as possible.."  Letter writer in Denver Post, after permanent change to DST (in U.S. Senate)  was announced (March 8, 2022).

Another jarring time change yesterday and more relentless stress on already stressed-out citizens, beaten down by unaffordability. The god awful, original bill, titled the Sunshine Protection Act, passed in a 308 to 117 vote, and still lies dormant - like a zombie ready come back.

We can't let it.  In fact, passing this bunkum to avoid changing clocks twice a year is like setting your home thermostat permanently on high so you don't have to change the settings twice a year.

And it's dumb for more ways than that example. An apt comment came from Democratic Rep. Mary Gay Scanlon of Pennsylvania, during the last Rules Committee meeting on the topic:

 "We all enjoy the extra hour or so of sunlight in the summer, but when people are considering this, they need to consider the extra hours of darkness in the winter." 

An even more apt comment comes from Beth Ann Malow M.D.,  Professor of Neurology and Pediatrics in the Sleep Disorders Center Of Vanderbilt University:

"It’s not one hour twice a year. It’s a misalignment of our biologic clocks for eight months of the year. When we talk about DST and the relationship to light we are talking about profound impacts on the biological clock, which is a structure rooted in the brain. It impacts brain functions such as sleep-wake patterns and daytime alertness”

DST was almost on the way out when a new meme struck which hailed the benefits of DST as an "energy saver" and my bull pocky  and PR radar instantly went off. 

Here's the other dirty secret exposed in the book, Spring Forward: The Annual Madness of  Daylight Saving Time'  ' by Michael Downing:

 "Daylight Saving increases gasoline consumption and it's not a small matter. The National Association of Convenience Stores, who sell 80 percent of our gasoline, says that the one month of extra daylight we got recently in 2005 was worth one billion dollars to them".

That is a staggering amount of gasoline and oil consumed! It confirms my suspicion that there were no real energy savings and in fact more energy waste.:  Also a large amount of carbon (almost 0.2gT) added to the atmosphere.

I've always felt DST or 'Daylight Saving Time' was a dumb notion. I mean you already have a natural clock cycle in standard time, based on actual Sun positions, so why mess with it? Well, because so a select few (like the convenience stores) can make profits via the false benefits claimed.

Leave out the energy aspect, DST is just a plain calamity for the nation's health - not to mention triggering drowsiness in drivers still not adapted because their circadian rhythms haven't yet adjusted. Indeed, a recent Stanford study, see link at the bottom, noted:

"Every spring, Americans dutifully adjust their clocks forward to daylight saving time, and every fall, back to standard time — but no one seems very happy about it. The biannual time shift is not only inconvenient, it’s also known to be acutely bad for our health. The collective loss of an hour of sleep on the second Sunday in March has been linked to more heart attacks and fatal traffic accidents in the ensuing days.  

Now, a study by Stanford researchers finds there are longer-term hazards and better alternatives. They compared three different time policies: permanent standard time, permanent daylight saving time and biannual shifting to see how these affected circadian rhythms and health across the country. (Circadian rhythm is the body's innate 24-hour cock which regulates physical processes). ....Indeed, by modeling light exposure, circadian impacts and health characteristics county by county, the researchers estimate that permanent standard time would prevent some 300,000 cases of stroke per year and result in 2.6 million fewer people having obesity

Prof. Beth Ann Malow of Vanderbilt offered her own thesis (back in 2019)  that the year in and year out switching times is bad for the brain, e.g

https://news.vumc.org/2019/11/04/malow-daylight-saving-time-brain-impact-commentary

She may have a point. There is also good evidence it significantly lowers productivity which the economists are always fretting over.  Thus, a survey conducted by the American Association of Sleep Medicine showed that  55 % of Americans reported feeling tired  after the transition to DST.  Further, "the group's health advisory says moving into and out of DST can adversely affect sleeping and waking patterns for 5- 7 days."

Well, think about it. Whereas before the DST switch you awakened at 7:15 a.m. to arrive at work by 8:00 a.m. that now becomes effectively a wake up time of 6:15 a.m. because the clock time is moved one hour ahead. Everything is shifted and you're trying to catch up to the change because of it, as well as having lost an hour of sleep.

Effect: Productivity crashes as medical problems rise and auto accidents proliferate. Result? Even more pronounced unaffordability (from medical debt) and grief because of a short-sighted non-solution to yearly clock changing.

See Also:

America tried permanent daylight saving time in the 1970s — and hated it

And:

What permanent daylight saving time would look like in your state

And:

Opinion | Daylight saving time is bad for your health. Here's a better option. - The Washington Post

Excerpt:

Daylight saving time will end this weekend, much to the dismay of people who dread ending their workday in darkness. Some dislike the change so much that they want to make DST permanent – as legislation that passed the House in July and has President Donald Trump’s support would do.

On the surface, this might sound like a great idea. Who wouldn’t want more afternoon sunlight? But it would be terrible for people’s health, which is why many major medical organizations oppose it, including the American Academy of Sleep Medicine, American Academy of Neurology and American Medical Association. Instead, they advocate keeping standard time in place year-round.

With year-round DST, many more Americans would be waking up in the dark and going to school and work before sunrise. That might benefit recreational activities done later in the day, which is why the golf industry is a prominent proponent of making DST permanent. But it goes against human’s natural circadian rhythms, which rely on morning light to set internal clocks and regulate sleep-wake cycles.

In fact, exposure to daylight later in the evening “tends to keep us awake and delay when we wake up the next morning,” said Erik Herzog, a circadian biology researcher at Washington University in St. Louis.

Advocates for permanent DST are not wrong to want to end the biannual time change, which substantially disrupts people’s circadian rhythms. Multiple studies have documented increases in car crashes, heart attacks and workplace accidents, including medical errors, in the days following the time change. The spring transition is particularly disruptive because people lose an hour of sleep, which explains why a 2025 poll found that just 12 percent of Americans favor the current system.

The problem is that year-round DST would further deprive people of morning sunlight, especially during the winter months, when days are already short.

WSJ Report: New Medicare Plans With More Benefits' Cuts - As Well As Cancellations, Part D Tier Changes

 The new Medicare enrollment period has begun and seniors are being advised to "move faster this year rather than waiting in the new enrollment period", i.e. to settle on a plan. This from the WSJ Personal Journal piece: 'New Medicare Plans Offer Fewer Benefits',  Oct. 19, p. A14). 

As this important article notes:

"Medicare enrollees have until Dec. 7 to select coverage for 2027. This year a lot of seniors will be looking for new coverage. Specifically, the 3.2 million people enrolled in Medicare Advantage plans that are closing or withdrawing  from their markets, according to 'Chapter' - a national Medicare brokerage."

The piece cites (quotes) a number of seniors who don't want more restrictive plans - which seems to be the norm now with Medicare Advantage.  So as the article advises:

"If you want the least restrictive option, you can choose traditional Medicare, but you will need a special supplemental policy-  known as a Medigap plan - to protect you from big, out of pocket charges."

Both Janice and I elected to go with traditional Medicare because we knew from friends how restrictive the MA plans were. You couldn't choose your own doctor, for example, and if one needed a special test or scan it was often a gamble whether it would be approved.  But Medicare Supplement plans are not cheap. Our total monthly outlays for our plans (we have separate ones) is nearly $600 - and often increases year to year.

Again, let's review the basics: The government pays Medicare Advantage insurers a set amount for each person who enrolls, with higher rates for sicker patients. And the insurers, among the largest and most prosperous American companies, from the get-go developed elaborate systems to make their patients appear as sick as possible, often without providing additional treatment.  See graphic (from NY Times, Oct. 9, 2022):

As a result, a program originally proposed to help lower health care spending has instead become a voracious financial gobbler. Now even more costly than the traditional government program it was meant to improve.  

A fall, 2022 Kiplinger Retirement Newsletter was clear on what is now happening to new Medicare Advantage beneficiaries: being denied access in record numbers.  Indeed, according to Kiplinger, at a rate nearly twice that before the pandemic struck in 2020.  Why?  Because the number of new Advantage beneficiaries has itself nearly doubled.  Naturally, from these early alarms one might expect the costs and restrictions of benefits to increase. And this is what's revealed in the WSJ piece, including there's always kind of a trade - you don't have to pay premiums like Janice and I do for our Medigap plans - but you will get hit elsewhere. 

As the article warns:

"Seniors will still have access to lots of Medicare Advantage plans with no premiums. The bite will come elsewhere, including that insurers will increase what you pay when you get care. According to analysts at Leerink Parters, Medicare Advantage plans are raising maximum out of pocket costs - including copays and other charges- by 10 percent on average."

Adding:

"Primary -care doctor visits are mostly still free, but some Medicare Advantage Plans are increasing copays for specialists or switching to coinsurance, which means you pay a share of the total cost rather than a flat amount."

So in the case of an MRI, for example, such as the one I had on May 5th to obtain a better resolution of the cancer lesions manifested in an earlier PSMA scan - you may have to cough up one third. If the MRI total cost was $13, 500 that means you will have to put in $4, 500 of your own.  (The same of course applies to the PET/PSMA scans or other special scans or treatments.)

As if this wasn't bad enough we also learn (ibid.):

"Large Medicare Advantage  insurers are also trimming many of the extra benefits that are popular with seniors - things like dental plans, coverage for over the counter health products and transportation."

The piece also warns of the 'tricky issues' in drug benefits. Included in these are:

- Stand alone drug plans continue to shrink, to 316 next year - a 55% fall from 2024.

- Many will face a substantial increase in their monthly bill if they choose to remain on the same Part D plan.

- Medicare Advantage drug deductibles are increasing by 30 percent on average.

- Some medications are being moved between tiers raising the costs for people who take them.

- For a small number of MA plans insurers are capping enrollment - meaning that once a certain number sign up-  the plan no longer is available.

In respect to the last the WSJ piece advises:

"If you find something you like but you're not ready to decide you might want to check with the insurer that it will be around later."

Good advice amidst a morass of Medicare Advantage confusion.

See Also:

WSJ Regular Pages Expose WSJ Editors' Claim of "Democratic Assault on Medicare Advantage" As Codswallop

And:

Healthcare Policy | Insurance

Here is the Truth: Medicare Advantage Is Neither Medicare Nor an Advantage

by Wendell Potter | January 15, 2023 - 7:23am | permalink

Excerpt:

Right now, well-funded lobbyists from big health insurance companies are leading a campaign on Capitol Hill to get Members of Congress and Senators of both parties to sign on to a letter designed to put them on the record “expressing strong support” for the scam that is Medicare Advantage.

But here is the truth: Medicare Advantage is neither Medicare nor an advantage.

And I should know. I am a former health-care executive who helped develop PR and marketing schemes to sell these private insurance plans.

And:

Healthcare Policy

The Simplicity of Medicare for All: Better, Cheaper, and Less Cruel

by Dean Baker | October 27, 2026 - 4:48am | permalink

— from Beat the Press

With soaring health care costs and a plunge in coverage rates, there is growing interest in Medicare for All. Polls show that a large majority of the public now favors a universal Medicare system, with even a substantial share of Republicans now backing it.

A universal Medicare system has a huge amount to recommend it. Of course, the main point is to ensure that people can get the health care that they need, but it also eliminates a vast amount of waste in the national health care system.

In addition to the $300 billion a year that we spend directly on operating the nation’s health insurance system, another $1.6 trillion is spent on administrative costs running our health care system by providers like hospitals and doctors’ offices, according to analysis by David Himmelstein, Terry Campbell, and Steffie Woolhandler. These additional costs are all the office personnel and various navigators involved in assisting patients in dealing with insurers. They found that in the United States we spend roughly 31 percent of each dollar in health care spending on administrative costs, whereas under Canada’s universal Medicare system, the cost is just 16 percent of spending.

» article continues...

And:

Stop the Medicare “Advantage” Scam Before Medicare is Dead

Thom Hartmann

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Corporate America | Healthcare Policy

by Thom Hartmann | August 6, 2022 - 6:08am

---

— from The Hartmann Report

Excerpt:

Congress must pass a law to stop the deceptive advertising of Medicare Advantage plans. Only Medicare should be able to call itself Medicare.

Unless you’ve been out of the country for the past few years, you’ve seen the ads on TV featuring Joe Namath, Jimmy Walker, or William Shatner hawking so-called “Medicare Advantage” plans.

Medicare Advantage is not Medicare.

It’s private health insurance being offered to people over 65, with the bill paid for by Medicare. Once you get on an Advantage plan it’s very difficult to get off, and if you’ve been on for more than a year you may not be able to go back to regular Medicare with a Medigap plan at all.

And:

How Medicare Advantage Scammers Get Away With It