What are the common causes of sinus pressure. How can you differentiate between allergies, viral infections, and bacterial sinusitis. What are effective relief strategies for sinus congestion. When should you seek medical attention for sinus symptoms.

Understanding Sinus Pressure: A Comprehensive Overview

Sinus pressure affects millions of Americans annually, causing discomfort and impacting daily life. This article delves into the causes, symptoms, and relief strategies for sinus-related issues, providing valuable insights for those seeking to understand and manage their condition.

Common Causes of Sinus Pressure: Allergies, Infections, and Environmental Factors

Sinus pressure can stem from various sources, each with its unique characteristics and treatment approaches. Let’s explore the primary culprits behind sinus discomfort:

• Seasonal and chronic allergies
• Viral infections
• Bacterial infections
• Environmental factors (e.g., humidity)

Understanding the root cause of your sinus pressure is crucial for effective treatment. While symptoms may overlap, certain indicators can help differentiate between allergies and infections.

Allergies: The Seasonal Sinus Saboteur

Allergic reactions, particularly hay fever, can trigger sinus pressure and related symptoms. How can you identify allergy-induced sinus issues?

• Clear, runny nasal discharge
• Itchy, watery eyes
• Symptoms tied to specific seasons or allergens
• Persistent sneezing and congestion

Viral vs. Bacterial Infections: Decoding the Difference

Both viral and bacterial infections can cause sinus pressure, but distinguishing between the two is essential for proper treatment. What are the key indicators of a bacterial sinus infection?

1. Symptoms persisting for more than 10 days
2. A “double worsening” pattern, where symptoms improve before worsening again
3. High fever (over 102°F)
4. Asymmetric pain in sinus areas

Recognizing Severe Sinus Symptoms: When to Seek Medical Attention

While most sinus issues resolve on their own or with minimal intervention, certain symptoms warrant immediate medical attention. What are the red flags that indicate a potentially serious sinus condition?

• Severe or constant headache
• Neck pain or stiffness
• Extreme sleepiness or changes in mental state
• Persistent chest pain or pressure
• Difficulty breathing
• Vision changes or eye pain
• Swelling or redness around the eyes, cheeks, or forehead

It’s important to note that some of these symptoms may overlap with severe COVID-19 infections. If you experience any of these signs, consult a healthcare professional promptly.

Effective Relief Strategies for Sinus Pressure and Congestion

Managing sinus pressure involves a combination of self-care techniques and, when necessary, medical interventions. What are some effective ways to alleviate sinus discomfort?

Home Remedies and Over-the-Counter Solutions

• Rest and hydration
• Over-the-counter pain relievers (e.g., acetaminophen, ibuprofen)
• Nasal irrigation using devices like neti pots
• Petroleum jelly for sore nostrils
• Cool-mist humidifiers to soothe airways
• Steam inhalation from hot showers
• Warm compresses on affected areas

Medical Interventions for Persistent Sinus Issues

When home remedies fail to provide relief, medical treatments may be necessary. What options are available for more severe or persistent sinus problems?

• Prescription antihistamines or decongestants for allergy-related sinus pressure
• Oral antibiotics for confirmed bacterial infections
• Corticosteroid nasal sprays to reduce inflammation
• Immunotherapy for chronic allergy sufferers

The Role of Telemedicine in Sinus Care: Convenient Access to Expert Advice

In today’s digital age, telemedicine offers a convenient solution for those seeking medical advice for sinus issues. How can telemedicine benefit patients with sinus concerns?

• 24/7 access to healthcare professionals
• Remote diagnosis and treatment recommendations
• Prescription services when necessary
• Guidance on differentiating between various sinus conditions

Telemedicine platforms like Teladoc provide patients with the opportunity to consult with experienced physicians from the comfort of their homes, ensuring timely and efficient care for sinus-related concerns.

Preventing Sinus Pressure: Proactive Measures for Long-Term Health

While treating sinus pressure is essential, prevention is equally important. What steps can individuals take to minimize their risk of developing sinus issues?

1. Maintain good hygiene practices, including regular hand washing
2. Use air purifiers to reduce allergens in the home
3. Stay hydrated to keep nasal passages moist
4. Avoid known allergens and irritants
5. Practice stress-reduction techniques, as stress can exacerbate sinus problems
6. Consider allergy testing to identify specific triggers
7. Use saline nasal sprays to keep nasal passages clear

The Impact of Sinus Pressure on Quality of Life: Addressing Chronic Issues

For some individuals, sinus pressure becomes a chronic issue, significantly impacting their daily lives. How can those with persistent sinus problems manage their condition and improve their quality of life?

Lifestyle Modifications for Chronic Sinus Sufferers

• Dietary changes to reduce inflammatory foods
• Regular exercise to boost immune function
• Sleeping with an elevated head to promote sinus drainage
• Using air filters and dehumidifiers in living spaces
• Exploring alternative therapies like acupuncture or aromatherapy

Long-Term Management Strategies

Chronic sinus issues may require ongoing management and collaboration with healthcare providers. What long-term strategies can help individuals cope with persistent sinus pressure?

• Developing a personalized treatment plan with a specialist
• Regular follow-ups to monitor progress and adjust treatments
• Considering surgical options for severe cases (e.g., septoplasty for deviated septum)
• Joining support groups or online communities for shared experiences and advice
• Keeping a symptom journal to identify patterns and triggers

The Future of Sinus Care: Emerging Treatments and Research

As medical science advances, new treatments and diagnostic tools for sinus-related issues continue to emerge. What innovations are on the horizon for sinus care?

Promising Developments in Sinus Treatment

• Biologics for chronic sinusitis and nasal polyps
• Advanced imaging techniques for more accurate diagnosis
• Minimally invasive surgical procedures
• Personalized medicine approaches based on genetic profiles
• Nasal microbiome research for targeted therapies

These advancements offer hope for more effective and personalized treatments for individuals suffering from chronic sinus pressure and related conditions.

Sinus Pressure in the Context of Global Health: COVID-19 Considerations

The COVID-19 pandemic has brought increased attention to respiratory symptoms, including those related to sinus pressure. How has the pandemic impacted the diagnosis and treatment of sinus issues?

Differentiating Sinus Symptoms from COVID-19

While some symptoms may overlap, certain characteristics can help distinguish between sinus issues and COVID-19 infections. What are the key differences to be aware of?

• Fever patterns (typically higher and more persistent in COVID-19)
• Presence of additional symptoms like loss of taste or smell
• Respiratory distress (more common in severe COVID-19 cases)
• Timeline of symptom progression

Adapting Sinus Care During a Pandemic

The healthcare landscape has evolved in response to COVID-19. How has sinus care adapted to these changes?

• Increased use of telemedicine for initial consultations
• Enhanced safety protocols for in-person examinations
• Greater emphasis on differential diagnosis
• Incorporation of COVID-19 testing when appropriate

As we navigate the complexities of sinus care in the context of global health challenges, staying informed and maintaining open communication with healthcare providers remains crucial for optimal management of sinus-related issues.

Sinus Pressure – When You Should Worry

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Are you battling a common cold, agitated by allergies or suffering from a sinus infection? Or, perhaps you’re wondering if your symptoms are a sign of  COVID-19. Sinus infections affect 28 million people in the U.S. every year.¹ Read on for a review of sinus symptoms and ways to get relief.

Sinus trouble: consider the causes

Not only can seasonal allergies or chronic allergies impact the sinuses, but humid air can cause a clogged or stuffy feeling in the nose. An infection—either brief or long-lasting—can also take hold.

Sinusitis ailments are not only a burden for allergy sufferers, they can be a challenge for doctors, too, especially as patients and doctors alike are on high alert for warning signs of a possible COVID-19 infection. Three of the most common causes of sinus symptoms are allergies, viral infections and bacterial infections. But these can be tough to tell apart because of overlapping symptoms.

The lowdown on stressed sinuses

A flare-up of seasonal allergies, called hay fever, causes stuffy noses and irritated sinuses. But allergy sufferers will notice mainly that they have a runny (and sometimes itchy) nose with clear output and itchy, watery eyes. Their symptoms are also often tied to certain times of year and specific allergens like animal dander, dust, pollen or mold.

If you’re plugged up with thick mucus that’s green or yellow, you could have an infection. Sinus infections—whether caused by bacteria or a virus—can also bring along other symptoms like mild headache, fatigue, weakness or a cough. Viruses are far more likely to be the cause of sinus infections. Certain symptoms increase the probably of bacterial sinusitis:

• Persistent sinusitis symptoms for longer than 10 days, especially with “double worsening.” This means symptoms start to improve and then get worse a few days later.
• A fever, especially a high one over 102 ℉.
• Asymmetric pain (one side much worse than the other) in one or more sinus areas. These include under or above the eyes and above the bridge of the nose.

“A bacterial infection could be serious if you’re having a severe or constant headache, neck pain or stiffness, extreme sleepiness or a change in mental state,” says physician Kyon Hood, president, Teladoc Health Medical Group. Some of these symptoms overlap with serious or worsening COVID-19 infections, so it’s important to keep a close eye on any symptoms that may signal low blood-oxygen levels, like persistent pain or pressure in the chest, trouble breathing, new confusion or pale, gray or bluish tone to the skin, lips or nail beds, depending on skin tone. “Also watch out for redness and swelling of the cheek, forehead or around the eyes; or double vision, decreased vision or pain when opening or moving your eyes,” he advises.

How Teladoc works

Teladoc gives you best-in-class care from doctors from wherever you are. Access our doctors by phone or video to diagnose, treat and even prescribe medicine if needed for common conditions like allergies, cough, flu, pink eye, rashes, sinus infections, sore throats, stomach bugs, UTI and more. Schedule a visit now to talk to a doctor 24/7 for non-emergency conditions.

Pain in the face for a few days in a row along with worsening symptoms may require treatment, Dr. Hood explains, but noted that determining the difference between a bacterial and viral infection isn’t always clear-cut.

How you can find relief

It’s important to review your sinusitis symptoms and how long you’ve been having them with a physician. Teladoc doctors can help you determine the possible cause of your symptoms, especially if you’re concerned you may have COVID-19. If your sinus problems are caused by allergies, there are strategies and medications that can help bring relief. While bothersome mucus is often an effect of a viral infection, a bacterial infection can develop after a virus has taken hold. In this case, oral antibiotics can help with healing.

Regardless of what’s causing all this congestion, try to put the to-do list aside and get some rest. If you’re having pain, you can take an over-the-counter (OTC) pain reliever like acetaminophen or ibuprofen. Petroleum jelly can soothe a sore nose on the outside, while nasal drops or a nasal irrigation device (like the neti pot) can help flush out nasal passages on the inside. “Treat and blow each nostril separately,” Dr. Hood says.

Also, don’t forget about all that mucus you’re losing: Moisture should play a key role in your return to good health. “Drink plenty of liquids like water, juice, warm broth or soup,” Dr. Hood says, “and use a cool-mist humidifier to soothe the airways, especially when sleeping.” Steam from hot showers can help reduce inflammation and make breathing easier. Also try a warm compress on the nose, cheekbones or forehead for relief from sinusitis symptoms, he adds.

Pseudoephedrine (also known as Sudafed), an OTC oral decongestant, can help relieve congestion, says Dr. Hood. He notes that certain people—such as those with high blood pressure—must use this medicine with caution. Dr. Hood advises against topical decongestants or nasal sprays such as oxymetazoline (also known as Afrin). “These can cause ‘rebound congestion,’ which actually increases the frequency of runny noses after the medication wears off,” he explains.

Get relief now

Remember that many sinus problems have overlapping symptoms, so it’s best to talk to a Teladoc physician about what’s causing the issue. You can reach out to Teladoc online or by app 24/7 to talk to one of our U.S. board-certified doctors.

Let’s get to the root of the illness so you don’t have to suffer with your sinuses any longer. If you think seasonal allergies may be the cause of your sinusitis symptoms, get fast relief from your allergies today.

Updated June 9, 2021

¹https://www.cdc.gov/nchs/fastats/sinuses.htm

This portion of the Teladoc Health website occasionally offers health, fitness, and nutritional information and is provided for educational purposes only. You cannot rely on any information provided here as a substitute for or replacement of professional medical advice, diagnosis or treatment. Teladoc Health cannot assure that the information contained on this site always includes the most recent findings or developments with respect to the particular subject matter covered.

If you ever have any concerns or questions about your health, you should always consult with a physician or other healthcare professional. Do not disregard, avoid or delay obtaining medical- or health-related advice from your healthcare professional because of something you may have read on this site. The use of any information provided on this site is solely at your own risk.

If you are in the United States and think you are having a medical or health emergency, call your healthcare professional, or 911, immediately.

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Sinusitis – Harvard Health

How to get relief without antibiotics

More than 20 million Americans will have at least one bout of sinusitis this year. Most will be uncomfortable, and many will miss work or school. Nearly all will recover from their sinus infections, but an unfortunate few may develop complications. If you understand sinusitis, you can reduce your chances of developing the problem and if sinusitis strikes, you’ll know how to speed your recovery and lower your risk of complications.

Your sinuses

Your sinuses are air-filled chambers located in the bones of your face; because they surround the nose, they are also known as the “paranasal sinuses.” Each of us has four pairs of sinuses (see figure).

Each of the sinuses is lined by a membrane that produces mucus. When you’re healthy, the mucus is a thin, watery fluid that flows freely from your sinuses into the upper part of your nose. But when your sinuses become inflamed, the mucus gets thick and sticky, so it can’t flow through the tiny openings, called ostia, that lead to the nose. Fluid builds up in the sinuses, causing pressure and pain, and just like that, you’ve got sinusitis.

What causes sinusitis?

Sinusitis is an infection most often caused by viruses or bacteria or both. Cold viruses and bacteria that get into our sinuses usually don’t cause trouble unless sinus drainage is blocked. Blockage of the narrow sinus draining channels is the main reason we get sinusitis, and restoration of drainage is the key to treatment.

What triggers sinusitis?

The common cold is the leading culprit. The average adult gets two to three colds a year, and the average child gets six to 10. Colds are caused by viruses, not bacteria, and antibiotics are useless for treatment. But cold viruses produce swelling of the nasal tissues, which can sometimes block the sinuses. Colds also change the mucus, preventing it from doing its normal job of clearing viruses and bacteria from the sinuses.

You may get some sinus pressure when you get a cold, but that doesn’t mean you have sinusitis or that you need an antibiotic. Only about one cold in 100 leads to sinusitis, and you can make the odds work for you by doing what it takes to keep your sinuses draining (see below). You can also help your cause by blowing your nose gently without pinching it tightly; forceful blowing can force bacteria up into your sinuses.

Many other things can block your sinuses and lead to infection. The list includes allergies, cigarette smoke and other irritating fumes, changes in barometric pressure during flying or scuba diving, nasal polyps, and a deviated nasal septum.

Sinus infection symptoms

Painful pressure is the main symptom of a sinus infection. Depending on which sinus is involved, the pain is in the forehead (frontal sinusitis), over the cheek or in the upper jaw and teeth (maxillary sinusitis), behind the eyes (ethmoid or sphenoid sinusitis), or at the top of the head (sphenoid sinusitis). Sinus pain usually increases when you bend forward.

Nasal congestion and a thick, dark-colored nasal discharge are also common during a sinus infection. When the mucus drips into your throat from the back of your nose, you’ll notice a foul taste, and you may get bad breath or a cough. You may temporarily lose your sense of smell or taste. Finally, you may feel feverish, achy, and tired.

Sinusitis diagnosis

In most cases, your doctor can diagnose sinusitis simply by asking about your symptoms. If pressing over your sinuses causes pain, sinusitis is likely. CT scans can be very helpful if your sinusitis is unusually severe or if your doctor suspects complications; plain x-rays are less useful. ENT (ear, nose, and throat) specialists can also diagnose sinusitis by looking up your nose with a scope.

Sinus infection treatment: Promoting drainage

Many people with sinusitis will recover quickly and completely without taking antibiotics simply by promoting drainage. Here’s what you should do:

• Drink lots of water. Good hydration helps keep the mucus thin and loose.
• Inhale steam. Linger in a hot shower. Boil a kettle, pour the water into a pan, and bend over the pan with a towel over your head to inhale the steam. Even hot tea or chicken soup will help; the secret ingredient is the steam. One way or another, inhale steam three to four times a day.
• Sleep with your head elevated. If your pain is only on one side, sleep with the pain-free side of your face on the pillow.
• Use decongestants. Tablets containing pseudoephedrine or phenylephrine are very helpful but may sometimes raise your blood pressure, speed your pulse, or make you jittery and keep you up at night. Nasal sprays containing phenylephrine or oxymetazoline don’t have these side effects, but if you use them too often or for too long, your nose can become irritated or dependent on them.
• Ask your doctor about prescription nasal sprays containing steroids, particularly if you have allergies or if your sinusitis is stubborn.
• Use a saline (salt water) nasal spray to loosen mucus and rinse your sinuses.
• Use antihistamines wisely. They’re great for allergies and when your nose waters from a cold, but they make mucus thick and hard to drain, the last thing you want in sinusitis.
• A warm compress on your face may soothe sinus pain. Over-the-counter pain relievers such as aspirin or acetaminophen will help reduce pain and fever.

Antibiotics for a severe sinus infection

You may be surprised that antibiotics are not listed as the first step in treatment. While many patients with sinusitis expect antibiotics, they aren’t usually needed if good drainage is achieved.

Antibiotics have potential disadvantages. They can trigger allergic reactions or cause side effects. Widespread use of antibiotics has encouraged the spread of antibiotic-resistant bacteria (“superbugs”). And many of these drugs are expensive.

Still, if your sinusitis does not improve with two to four days of drainage therapy, or if it’s very severe to begin with, it probably needs an antibiotic to help get rid of the trapped bacteria. Antibiotic therapy for just three to seven days is generally as effective as traditional 10- to 14-day treatment for uncomplicated acute sinusitis.

Many bacteria can cause acute sinusitis; the most common include some with fearsome names like Pneumococcus, Streptococcus, Hemophilus, and Moraxella. Unless you have a sinus puncture (an invasive test that’s rarely performed by ENT specialists unless unusual organisms or complications are suspected), there’s really no way to know which bacteria are causing your sinusitis. Cultures of your mucus or your nose,  even if they are obtained through a nasal speculum, are not helpful because they are always contaminated by the many bacteria that live in every nose.

With these considerations in mind, many infectious disease specialists recommend initial sinusitis treatment with one of the old standbys, such as trimethoprim-sulfamethoxazole (a combination that includes a sulfa drug), amoxicillin (a type of penicillin), or doxycycline (a type of tetracycline). If these fail “” or if the patient is unusually ill to begin with “” doctors may turn to amoxicillin-clavulanic acid or a quinolone (such as levofloxacin).

Complications

The sinuses are surrounded by critical structures, including the brain, the eyes, and the skull. On rare occasions, sinus infections can spread to one of these areas. Notify your doctor promptly if your sinusitis gets worse with one or more of these warning symptoms:

• High fever
• Severe headache
• Mental confusion or stiff neck
• Swelling of the cheek, forehead, or roof of the mouth
• A swollen, red, painful eye
• Impaired vision
• Difficulty breathing, swallowing, or speaking.

Fortunately, none of these problems is likely. Still, they serve to remind us that sinusitis is not just the sniffles. Patients with weakened immune systems always require careful medical evaluation and treatment for sinusitis.

Chronic sinusitis

Sinusitis that lasts more than three weeks or recurs more than three times a year is called chronic sinusitis. The most common cause of chronic sinusitis is inadequately treated acute sinusitis, and since the diagnosis and treatment of acute sinusitis have improved so much, chronic sinusitis is less common than it used to be.

Most patients with chronic sinusitis will benefit from an ENT evaluation, which usually includes looking inside both nostrils with a scope. The doctor may also order a CT scan. That’s because anatomic problems such as nasal polyps or a deviated septum are often responsible. Since allergies are also present in many cases, you may want to pursue formal allergy testing.

Persistent nasal discharge and congestion are the main symptoms of chronic sinusitis. Unless flares of acute sinusitis develop, headache and fever are minimal.

In addition to the bacteria that cause acute sinusitis, Staphylococcus, anaerobic bacteria, and fungi may lead to chronic sinusitis. Antibiotics can be helpful, especially for flares, but they are less important than nasal irrigation, decongestants, and prescription nasal spray. If allergies are present, antihistamines can help. In severe cases, oral steroids may be necessary. As with acute sinusitis, the key to successful treatment is drainage. In the case of chronic sinusitis, though, surgery may be needed. Your ENT doctor may remove nasal polyps, straighten a deviated septum, or use endoscopic surgery to open a good drainage channel between the sinus and your nose.

Living with your sinuses

It’s nice to know that CT scans, endoscopy, and surgery are available to evaluate and treat complicated acute sinusitis and stubborn chronic sinusitis. Fortunately, garden-variety acute sinusitis will respond to much simpler treatments.

To protect your sinuses, stay well-hydrated. Avoid tobacco smoke and irritating fumes. If you have allergies, try to avoid the things that trigger symptoms. Do your best to prevent catching a cold by washing your hands properly and keeping your distance from cold sufferers. When you catch a cold, blow your nose properly to keep bacteria from flooding into your sinuses. Treat symptoms of sinusitis promptly with steam inhalation, decongestants, and nasal irrigation. If you don’t recover as expected, or if you have serious sinusitis or warning symptoms, see your doctor for antibiotics and, possibly, nasal steroids.

Sine theorem. Formulas and evidence

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Railroad tracks are parallel to each other, and trees grow at an angle to the ground. Alas, with the ratio of the sides in a triangle, everything is not so simple: to determine them, you need the sine theorem.

Proof of the sine theorem

The sine theorem is as follows: the sides of a triangle are proportional to the sines of the opposite angles.

Draw a standard triangle and write the theorem with the formula:

Formula of the sine theorem:

Prove the theorem using the triangle area formula in terms of the sine of its angle.

From this formula we get two ratios:

1. On b we reduce, we use the rule of proportion and get:

From these two relations we get:

The sine theorem for the triangle is proved.

This theorem is useful to find:

• The sides of a triangle given two angles and one side.

• Angles of a triangle, given two sides and one included angle.

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Proof of the corollary of the sine theorem

The sine theorem has an important consequence. Let’s draw a triangle, describe a circle around it and consider the consequence through the radius.

, where R is the radius of the circle circumscribed about the triangle.

This is how three formulas for the radius of the circumscribed circle were formed:

The main meaning of the corollary of the sine theorem lies in this formula:

The doubled radius of the circumscribed circle is equal to the ratio of the side of the triangle to the sine of the opposite angle.

To prove the corollary of the sine theorem, consider three cases.

1. The angle ∠A = α is acute in triangle ABC.

Draw diameter BA ₁ . In this case, point A and point A ₁ lie in the same half-plane from the line BC.

Using the inscribed angle theorem, we see that ∠А = ∠А ₁ = α. The triangle BA ₁ C is right-angled, in it ∠ BCA ₁ = 90°, since it rests on the diameter BA ₁ .

To find leg a in triangle BA ₁ C, multiply the hypotenuse BA ₁ by the sine of the opposite angle.

BA ₁ = 2R, where R is the radius of the circle

Therefore:

For an acute triangle with a circumscribed circle, the theorem is proved.

2. The angle ∠A = α is obtuse in triangle ABC.

Draw the diameter of the circle BA ₁ . Points A and A ₁ on opposite sides of the straight line BC. Quadrilateral ACA ₁ B is inscribed in a circle, and its main property is that the sum of opposite angles is 180°.

Therefore, ∠А ₁ = 180° – α.

Recall the property of a quadrilateral inscribed in a circle:

It is also known that sin(180° – α) = sinα.

In triangle BCA ₁ , the vertex angle C is 90° because it rests on the diameter. Therefore, we find the leg and in this way:

α \u003d 2R sin (180 ° – α) \u003d 2R sinα

Therefore:

For an obtuse triangle with a circumscribed circle, the theorem is proved.

Commonly used obtuse corners:

3. Angle ∠A = 90°.

In rectangle ABC, angle A is right, and the opposite side is BC = α = 2R, where R is the radius of the circumscribed circle.

Therefore:

For a right triangle with a circumscribed circle, the theorem is proved.

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Examples of problem solving

The sine theorem and its consequences are actively used in problem solving. Let’s look at a few examples to reinforce the material.

Example 1. In triangle ABC ∠A = 45°, ∠C = 15°, BC = 4√6. Find AC.

How to solve:

• According to the triangle sum of angles theorem:

  ∠A + ∠B + ∠C = 180°

  ∠B = 180° – 45° – 15° = 120°

• Find the AC side using the sine theorem:

Answer: AC = 12.

Example 2. The hypotenuse and one of the legs of a right triangle are 10 and 8 cm, respectively. Find the angle opposite the given leg.

How we solve:

Let’s take the unknown angle as x. Then the aspect ratio looks like this:

Therefore:

So .

Answer: The angle is approximately 53.1°.

Memorize

The usual theorem: The sides of a triangle are proportional to the sines of the opposite angles.

Extended theorem: in an arbitrary triangle, the following relation holds:

, where R is the radius of the circumscribed circle around the triangle.

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Sine theorem with examples: classic and extended0003

When preparing for the exam in mathematics , an eleventh grader must remember a basic set of formulas that will help solve problems. One of them is the sine theorem, which reflects the relationship between the sides and angles of a triangle.

Recall that the proof of the theorem does not need to be taught, since the exam is focused on testing practical skills. It is better to devote time to analyzing examples in which the specified mathematical pattern can be applied.

Sine theorem with examples

Mankind has been familiar with the sine theorem for a long time – at the beginning of the 21st century, its proof was given in his work “The Book of the Unknown Arcs of the Sphere” by the Western Arab astronomer and mathematician Ibn Muaz al-Jayani.

There are two versions of the sine theorem:

• ordinary – establishes the relationship between the sides of a triangle and the sines of its angles;
• extended – relates the aspect ratio of a triangle to the radii of the circumscribed circle.

Statement of the usual sine theorem: the ratio of the sides of a triangle to the sines of the opposite angles are equal or the sides are proportional to the sines of the opposite angles.

Sine theorem with examples: classic and extended

Example 1. In a triangle ABC, side AB is 5 cm, and the sine of the opposite angle ACB = 3/5. Find side BC if the sine of angle CAB adjacent to side AB is 1/2.

Solution

Compose the ratio of the sides appearing in the condition and the sines of their angles:

AB : sin ∠ACB = BC : sin ∠CAB.

Substitute the known values:

5 : 3/5 = BC : 1/2.

Let’s express BC from this expression:

BC = (5 : 3/5) : 1/2 = 5 : 1/2 = 10 cm

Answer: BC = 10 cm

In triangle ABC, side AB is 10 cm and the opposite angle DAB is 30°. Find the remaining sides if the angle CAB is 60°.

Solution

To solve this problem, we will use the attached table, which shows the values ​​of the sines of the main angles. The rest of the solution will be similar to the previous example, with the exception of one small move. First, let’s make the ratio of the sides and sines of the opposite angles:

AB : sin ∠ACB = BC : sin ∠CAB = AC : sin ∠BAC.

At the first stage, we know only three of the six terms of this equality, and two of them are in indirect form:

10 : sin 30° = BC : sin 60° = AC : sin ∠BAC

If we remember that the sum of the angles of a triangle is 180°, then it is easy to find the remaining angle:

∠BAC = 180° – (∠ACB + ∠CAB) = 180° – (30° + 60°) = 90°.

We already know the third angle, so the equation will look like this:

10 : sin 30° = BC : sin 60° = AC : sin 90°.

Then we proceed as in the previous problem, expressing the sides in terms of the known terms of the expressions:

BC = sin 60° ∙ 10 : sin 30°,

Let’s turn to the table above and select from it the corresponding sines of known angles:

BC = √3/2∙ 10 : 1/2 = 10√3 cm see

Answer: BC = 10√3 cm; AC = 20 cm.

Sine theorem with examples: classic and extended

Extended sine theorem with examples

Statement of the extended sine theorem: the ratio of the sides of a triangle to the sines of the opposite angles are equal to each other and twice the radius of the circumscribed circle.

Example 3. Find the area of ​​a triangle if the diameter of the circumscribed circle D is 20 cm. Angle DAB = 30° and angle CAB = 60°.

Solution

To solve, we use the extended formulation of the sine theorem:

AB : sin ∠ACB = BC : sin ∠CAB = AC : sin ∠BAC = 2R.

In this formulation, we know two of the seven components, and we can determine another from basic knowledge of geometry:

• R = ½ D, therefore 2 R = D = 20 cm;
• ∠BAC = 180° – (∠ACB + ∠CAB) = 180° – (30° + 60°) = 90°.

Substitute the known values ​​into the original expression and get the ratio:

AB : sin 30° = BC : sin 60° = AC : sin 90° = 20 , but twice the radius of the circumscribed circle is known. This allows you to write expressions for finding all sides of the triangle:

BC = 20 ∙ sin 60°

AC = 20 ∙ sin 90°,

AB = 20 ∙ sin 30°.

Select from the table the values ​​of the sines of the angles and subtract the sides of the triangle:

BC = 20 ∙ sin 60° = 20 ∙ √3/2 = 10√3 cm see,

AB = 20 ∙ sin 30° = 20 ∙ 1/2 = 10 cm. Now it remains to find its area. To do this, we take the standard formula for the area of ​​​​an arbitrary triangle, which is equal to half the product of the sides and the sine of the angle between them

S = ½ ∙ a ∙ b ∙ sin α

Since we know all sides and all angles, we can choose any of them. Let’s take the sides AC and AB, as well as the angle CAB between them:

Note: the attentive reader noticed that our triangle is right-angled, since one of its angles is equal to 90°. In this case, you can do without knowing the sine of the angle by calculating the area of ​​​​the triangle as half the area of ​​\u200b\u200bthe rectangle, the length and width of which are equal to the legs of the triangle.

S = ½ ∙ BC ∙AB = ½ ∙ 10√3 ∙ 10 = 50√3 cm 903 96 2 .

Answer: S = 50√3 cm ² .

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Vladislav Baryshnikov

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900 02 Graduated from the Moscow Institute of Physics and Technology (Phystech) majoring in applied physics and mathematics.