What are the common causes of sinus pain and congestion. How can you differentiate between viral, bacterial, and allergic sinus infections. When should you seek medical attention for sinus-related symptoms. What are effective home remedies for sinus congestion.

The Nature of Sinus Pain and Congestion

Sinus pain and congestion are common ailments that affect many individuals, particularly during cold and allergy seasons. These conditions can cause significant discomfort and impact daily life. To better understand and manage these symptoms, it’s essential to delve into their nature, causes, and potential treatments.

What are sinuses?

Sinuses are air-filled cavities located in the facial bones surrounding the nose. They play a crucial role in filtering, warming, and humidifying the air we breathe. When these cavities become inflamed or blocked, it can lead to the symptoms we associate with sinus problems.

Identifying Sinus Pain and Congestion Symptoms

Recognizing the symptoms of sinus issues is the first step toward proper management and treatment. Common indicators include:

• Fullness, pressure, or pain on the face over a sinus
• Pain above the eyebrow, behind the eye, or under the cheekbone
• Blocked nose or nasal discharge
• Postnasal drip
• Swelling around one eye (in some cases)
• Bad breath or mouth breathing (less common)
• Sore throat and throat clearing due to postnasal drip

It’s worth noting that sinus pain is not a common symptom in children under 5 years of age.

Understanding the Causes of Sinus Congestion

Sinus congestion can stem from various sources, each requiring different approaches to treatment. The three primary causes are:

1. Viral Sinus Infection

Often part of the common cold, a viral sinus infection affects the lining of the nose and all the sinuses. This is the most common cause of sinus congestion.

2. Bacterial Sinus Infection

Occurring in about 5% of colds, a bacterial sinus infection typically starts as a viral infection but progresses when bacteria invade the sinuses. Symptoms may include increased sinus pain, return of fever, or redness and swelling around the eyelids or cheeks.

3. Allergic Sinus Reaction

Allergies, such as those triggered by pollen, can cause sinus congestion. Additional symptoms like sneezing, an itchy nose, and clear nasal discharge often accompany this type of reaction.

Diagnosing Different Types of Sinus Infections

Distinguishing between viral, bacterial, and allergic sinus infections is crucial for determining the appropriate treatment. Here are some key indicators:

How can you tell if a sinus infection is viral or bacterial?

While both viral and bacterial infections can cause colored nasal discharge, a bacterial infection is more likely if:

• The discharge becomes thick, resembling pus
• Sinus pain (not just congestion) is present
• There’s swelling or redness over any sinus
• Fever lasts more than 3 days or returns after being gone for over 24 hours
• Nasal discharge and post-nasal drip persist for over 14 days without improvement

What are the signs of an allergic sinus reaction?

Allergic sinus reactions typically present with:

• Sneezing
• Itchy nose
• Clear nasal discharge
• Sinus congestion

Treatment Options for Sinus Congestion

The treatment approach varies depending on the underlying cause of the sinus congestion:

Viral Sinus Infection Treatment

For viral infections, the primary treatment involves:

• Nasal washes with saline solution
• Rest and hydration
• Over-the-counter pain relievers if needed

Antibiotics are not effective against viral infections.

Bacterial Sinus Infection Treatment

Bacterial sinus infections typically require:

• Oral antibiotics prescribed by a healthcare provider
• Nasal saline washes to help clear congestion
• Pain relievers as needed

Allergic Sinus Reaction Treatment

For allergy-induced sinus congestion, treatment may include:

• Antihistamines
• Nasal corticosteroid sprays
• Decongestants (short-term use)
• Allergy shots in some cases

Home Remedies for Sinus Congestion

While medical treatments are often necessary, several home remedies can provide relief from sinus congestion:

1. Steam inhalation: Inhaling steam from a bowl of hot water or a hot shower can help moisturize the nasal passages and loosen mucus.
2. Nasal irrigation: Using a neti pot or saline spray can help flush out mucus and allergens from the nasal passages.
3. Hydration: Drinking plenty of water helps thin mucus and promotes drainage.
4. Warm compress: Applying a warm, damp towel to the face can help relieve pain and pressure.
5. Elevating the head: Sleeping with the head elevated can facilitate sinus drainage.

When to Seek Medical Attention for Sinus Symptoms

While many cases of sinus congestion can be managed at home, certain symptoms warrant immediate medical attention:

When should you call 911 for sinus-related symptoms?

• Severe trouble breathing (struggling for each breath, can barely speak or cry)
• Not moving or too weak to stand
• Any signs of a life-threatening emergency

When should you seek urgent care for sinus issues?

• Trouble breathing that doesn’t improve after clearing the nose
• Redness or swelling on the cheek, forehead, or around the eye
• Severe headache that’s getting worse
• Severe pain not relieved by over-the-counter medications
• Fever over 104° F (40° C)
• Signs of severe illness or distress

When should you contact your doctor about sinus symptoms?

Consider contacting your healthcare provider if:

• Headache lasts more than 48 hours
• Fever lasts more than 3 days or returns after being gone for over 24 hours
• Sinus pain (not just pressure) lasts more than 24 hours despite using nasal washes
• Thick yellow or green discharge doesn’t improve with nasal washes
• Sinus congestion and fullness last more than 14 days
• Nasal discharge persists for more than 2 weeks

Prevention Strategies for Sinus Congestion

While it’s not always possible to prevent sinus congestion, certain measures can reduce your risk:

How can you minimize the risk of sinus infections?

• Practice good hand hygiene to reduce exposure to viruses and bacteria
• Avoid known allergens if you have allergies
• Use a humidifier to keep nasal passages moist
• Avoid smoking and secondhand smoke
• Manage allergies and asthma effectively
• Stay hydrated to keep mucus thin

By implementing these preventive measures, you can potentially reduce the frequency and severity of sinus congestion episodes.

Understanding the Impact of Sinus Issues on Quality of Life

Sinus pain and congestion can significantly affect an individual’s daily life, impacting sleep, work performance, and overall well-being. Chronic sinus issues may lead to:

• Fatigue due to poor sleep quality
• Difficulty concentrating
• Reduced sense of smell and taste
• Mood changes, including irritability or depression
• Social isolation due to symptoms

How can you manage the impact of sinus issues on daily life?

To minimize the impact of sinus problems on your quality of life, consider the following strategies:

1. Maintain a consistent sleep schedule and create a sleep-friendly environment
2. Practice stress-reduction techniques like meditation or yoga
3. Communicate with employers or teachers about your condition if it affects performance
4. Stay connected with friends and family, even when symptoms are bothersome
5. Consider joining a support group for individuals with chronic sinus issues

By addressing both the physical symptoms and the broader impact on daily life, individuals can better manage sinus issues and maintain overall well-being.

Emerging Treatments and Research in Sinus Care

The field of sinus care is continuously evolving, with new treatments and research offering hope for those suffering from chronic sinus issues. Some promising areas include:

What are some innovative approaches to treating sinus problems?

• Balloon sinuplasty: A minimally invasive procedure to open blocked sinus passages
• Immunotherapy: Customized treatments to reduce allergic reactions
• Probiotics: Exploring the role of beneficial bacteria in sinus health
• Nasal microbiome research: Studying the balance of microorganisms in the nasal passages
• Advanced imaging techniques: Improving diagnosis and treatment planning

As research progresses, new treatments may become available, offering more effective and personalized approaches to managing sinus conditions.

The Role of Diet and Lifestyle in Sinus Health

While often overlooked, diet and lifestyle factors can play a significant role in sinus health. Certain foods and habits may exacerbate sinus issues, while others can potentially alleviate symptoms.

How can diet influence sinus health?

Consider the following dietary factors:

• Stay hydrated: Proper hydration helps keep mucus thin and easier to clear
• Reduce inflammatory foods: Some people find relief by limiting dairy, sugar, and processed foods
• Increase anti-inflammatory foods: Incorporate omega-3 rich foods, fruits, and vegetables
• Consider probiotics: Some studies suggest probiotics may support sinus health
• Be aware of food allergies: Undiagnosed food sensitivities could contribute to sinus issues

What lifestyle changes can support sinus health?

In addition to diet, consider these lifestyle modifications:

1. Regular exercise: Promotes overall health and can help improve sinus drainage
2. Stress management: Chronic stress can impact immune function and exacerbate sinus problems
3. Proper sleep: Adequate rest supports immune function and overall health
4. Air quality: Use air purifiers and keep living spaces clean to reduce allergens
5. Nasal hygiene: Practice regular nasal rinsing to clear irritants and maintain moisture

By addressing both diet and lifestyle factors, individuals may find additional relief from sinus issues and improve overall sinus health.

Navigating Sinus Issues in Special Populations

Sinus problems can affect people of all ages, but certain populations may face unique challenges or require special considerations in their care.

How do sinus issues differ in children?

Children’s sinus systems are still developing, which can impact how sinus problems present and are treated:

• Sinus pain is uncommon in children under 5 years old
• Children may be more prone to viral upper respiratory infections
• Allergies may develop or become apparent in childhood
• Treatment approaches may need to be adjusted for age and weight

What special considerations exist for older adults with sinus issues?

Older adults may face additional challenges when dealing with sinus problems:

• Increased risk of complications due to weakened immune systems
• Potential interactions between sinus treatments and other medications
• Chronic conditions may complicate sinus issues or their treatment
• Changes in nasal structure with age may impact sinus function

How do sinus problems affect individuals with compromised immune systems?

People with weakened immune systems require special attention when managing sinus issues:

• Higher risk of developing bacterial sinus infections
• May require more aggressive treatment approaches
• Need for careful monitoring to prevent complications
• Importance of preventive measures to reduce infection risk

Understanding these unique considerations can help ensure appropriate care for individuals in different life stages or with specific health conditions.

Sinus Pain or Congestion

Is this your child’s symptom?

• Fullness, pressure or pain on the face over a sinus
• Sinus pain occurs above the eyebrow, behind the eye, and under the cheekbone
• Other common symptoms can be a blocked nose, nasal discharge, or postnasal drip

Symptoms

• Most often, the pain or pressure is just on one side of the face.
• Swelling around just one eye.
• Other common symptoms are a stuffy or blocked nose or nasal discharge. Your child may also have a nasal drip down the back of the throat. This is called a postnasal drip.
• Less common symptoms are bad breath or mouth breathing. Also, may have a sore throat and throat clearing from postnasal drip.
• Age Limit. Sinus pain is not a common symptom before 5 years of age.

Causes of Sinus Congestion

• Viral Sinus Infection. Part of the common cold. A cold infects the lining of the nose. It also involves the lining of all the sinuses.
• Bacterial Sinus Infection. A problem when the sinus becomes infected with bacteria. (Occurs in 5% of colds). It starts as a viral sinus infection. Main symptoms are increased sinus pain or return of fever. The skin around the eyelids or cheeks may become red or swollen. Thick nasal secretions that last over 14 days may point to a sinus infection. This can occur in younger children.
• Allergic Sinus Reaction. Sinus congestion often occurs with nasal allergies (such as from pollen). Sneezing, itchy nose and clear nasal discharge point to this cause.

Treatment of Sinus Congestion

• Viral Sinus Infection. Nasal washes with saline. Antibiotics are not helpful.
• Bacterial Sinus Infection. Antibiotics by mouth.
• Allergic Sinus Reaction. Treatment of the nasal allergy with allergy medicines also often helps the sinus symptoms.
• All Thick Nasal Drainage. Nasal secretions need treatment with nasal saline when they block the nose. Also, treat if they make breathing through the nose hard. If breathing is noisy, it may mean the dried mucus is farther back. Nasal saline rinses can remove it.

Color of Nasal Discharge with Colds

• The nasal discharge changes color during different stages of a cold. This is normal.
• It starts as a clear discharge and later becomes cloudy.
• Sometimes it becomes yellow or green colored for a few days. This is still normal.
• Colored discharge is common after sleep, with allergy medicines or with low humidity. Reason: all of these events decrease the amount of normal nasal secretions.

Bacterial Sinus Infections: When to Suspect

• Yellow or green nasal discharge is seen with both viral and bacterial sinus infections. Suspect a bacterial infection if the discharge becomes thick (like pus). But, it also needs one or more of these symptoms:
• Sinus Pain, not just normal sinus congestion. Pain occurs mainly behind the cheekbone or eye or
• Swelling or redness of the skin over any sinus or
• Fever lasts more than 3 days or
• Fever returns after it’s been gone for over 24 hours or
• Nasal discharge and post-nasal drip lasts over 14 days without improvement

When to Call for Sinus Pain or Congestion

Call 911 Now

• Not moving or too weak to stand
• Severe trouble breathing (struggling for each breath, can barely speak or cry)
• You think your child has a life-threatening emergency

Call Doctor or Seek Care Now

• Trouble breathing, but not severe. Exception: gone after cleaning out the nose.
• Redness or swelling on the cheek, forehead or around the eye
• Severe headache and getting worse
• Severe pain and not better after using care advice
• Weak immune system. Examples are: sickle cell disease, HIV, cancer, organ transplant, taking oral steroids.
• Fever over 104° F (40° C)
• Your child looks or acts very sick
• You think your child needs to be seen, and the problem is urgent

Contact Doctor Within 24 Hours

• Headache lasts more than 48 hours
• Fever lasts more than 3 days
• Fever returns after being gone more than 24 hours
• Earache occurs
• Sinus pain (not just pressure) and fever
• You think your child needs to be seen, but the problem is not urgent

Contact Doctor During Office Hours

• Sinus pain (not just pressure or fullness) lasts more than 24 hours, after using nasal washes
• Thick yellow or green pus draining from nose and not improved by nasal washes. Exception: yellow or green tinged secretions are normal.
• Sinus congestion and fullness lasts more than 14 days
• Nasal discharge lasts more than 2 weeks
• You have other questions or concerns

Self Care at Home

• Normal sinus congestion as part of a cold

Seattle Children’s Urgent Care Locations

If your child’s illness or injury is life-threatening, call 911.

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Care Advice for Sinus Congestion

1. What You Should Know About Sinus Congestion:
   • Sinus congestion is a normal part of a cold.
   • Nasal discharge normally changes color during different stages of a cold. It starts as clear, then cloudy, turns yellow-green tinged, then dries up.
   • Yellow or green-tinged discharge. This is more common with sleep, antihistamines or low humidity. (Reason: decrease the amount of normal nasal secretions.)
   • Usually, nasal washes can prevent a bacterial sinus infection.
   • Antibiotics are not helpful for the sinus congestion that occurs with colds.
   • Here is some care advice that should help.
2. Nasal Saline to Open a Blocked Nose:
   • Use saline (salt water) nose spray (such as store brand). This helps to loosen up the dried mucus. If you don’t have saline, you can use a few drops of water. Use bottled water, distilled water or boiled tap water. Teens can just splash a little water in the nose and then blow.
   • Step 1: Put 3 drops in each nostril.
   • Step 2: Blow each nostril out while closing off the other nostril. Then, do the other side.
   • Step 3: Repeat nose drops and blowing until the discharge is clear.
   • How often: Do saline rinses when your child can’t breathe through the nose.
   • Saline nose drops or spray can be bought in any drugstore. No prescription is needed.
   • Reason for nose drops: Suction or blowing alone can’t remove dried or sticky mucus.
   • Other option: use a warm shower to loosen mucus. Breathe in the moist air, then blow each nostril.
3. Fluids – Offer More:
   • Try to get your child to drink lots of fluids.
   • Goal: Keep your child well hydrated.
   • It also will thin out the mucus discharge from the nose.
   • It also loosens up any phlegm in the lungs. Then it’s easier to cough up.
4. Humidifier:
   • If the air in your home is dry, use a humidifier. Reason: Dry air makes nasal mucus thicker.
5. Decongestant Nose Spray (Age 12 years or Older):
   • Use this only if the sinus still seems blocked up after nasal washes. Use the long-acting type (such as Afrin).
   • Dose: 1 spray on each side. Do this 2 times per day.
   • Always clean out the nose with saline before using.
   • Use for 1 day. After that, use only for symptoms.
   • Don’t use for more than 3 days. (Reason: Can cause rebound congestion).
   • Decongestants given by mouth (such as Sudafed) are another choice. They can also open a stuffy nose and ears. Side effects: They may make a person feel nervous or dizzy. Follow the package directions.
6. Pain Medicine:
   • To help with the pain, give an acetaminophen product (such as Tylenol).
   • Another choice is an ibuprofen product (such as Advil).
   • Use as needed.
   • Try saline first. Sometimes it alone relieves the pain.
7. Cold Pack for Pain:
   • For pain or swelling, use a cold pack. You can also use ice wrapped in a wet cloth.
   • Put it over the sinus for 20 minutes.
   • Caution: Avoid frostbite.
8. Allergy Medicine:
   • If the child also has nasal allergies, give an allergy medicine.
   • Long-acting allergy medicines (such as Zyrtec) are best. Reason: these meds do not cause your child to act sleepy. Age limit: 2 and older.
   • A single dose of Benadryl can be given for any breakthrough symptoms.
   • No prescription is needed. Age limit: 1 and older.
9. What to Expect:
   • With this advice, the viral sinus blockage goes away in 7 to 14 days.
   • The main problem is a sinus infection from bacteria. This can occur if bacteria multiply within the blocked sinus. This leads to a fever and increased pain. It needs antibiotics. Once on treatment, the symptoms will improve in a few days.
10. Return to School:
    • Sinus infections cannot be spread to others.
    • Your child can return to school after the fever is gone. Your child should feel well enough to join in normal activities.
11. Call Your Doctor If:
    • Sinus pain lasts more than 24 hours after starting treatment
    • Sinus congestion lasts more than 2 weeks
    • Fever lasts more than 3 days
    • You think your child needs to be seen
    • Your child becomes worse

And remember, contact your doctor if your child develops any of the ‘Call Your Doctor’ symptoms.

Disclaimer: this health information is for educational purposes only. You, the reader, assume full responsibility for how you choose to use it.

Last Reviewed: 07/17/2023

Last Revised: 12/30/2022

Copyright 2000-2023. Schmitt Pediatric Guidelines LLC.

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.

COVID-19, Health Talk, Sinus

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Sine theorem. Formulas and Proofs

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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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Grade 9 is a time full of new knowledge. In order not to get confused in the geometry theory, we recommend making cards with information on each topic. In this article you will find the most important thing about the cosine theorem.

Formulation and proof of the cosine theorem

First, let’s recall the Pythagorean theorem: in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

Formula Pythagoras :

a ² > + b ² > = c ² >, where a, b are legs, c is the hypotenuse.

In proof of the cosine theorem we use the formula for the length of a segment in coordinates. Consider this formula:

BC ² = (x ₂ – x ₁ ) ² + (y ₂ – y ₁ ) ²

In the proof of the BC cosine theorem is the side of the triangle ABC, which is denoted by the letter a. We introduce a convenient coordinate system and find the coordinates of the points we need. Point B has coordinates (c; 0).
Coordinates of point С — (b cos α; b sin α) for α ∈ (0° ; 180°).

cos ² α + sin ² α = 1 – basic trigonometric identity.

BC ² = a ² = (b cos α – c) ² + b ² sin ² α = b ² cos ² α + b ² sin ² α – 2bc cos α + c ² = b ² (cos ² α + sin ² α) – 2bc cos α + c ²

What was required to be proved.

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Using the law of cosines, you can find the cosine of the angle of a triangle:

• 0285 > 0, the angle α will be acute.
• When b ² + c ² – a ² = 0, angle α will be right.
• When b ² + c ² – a ² < 0, the angle α will be obtuse.

Memorize

When the angle α is straight, then the cosine theorem turns into the Pythagorean theorem.

Let’s formulate another proof of the cosine theorem .

Let us be given a triangle ABC in which height CD is lowered from vertex C to side AB. This means:

• AD = b × cos α,
• DB = c – b × cos α.

Write the Pythagorean theorem for two right triangles ADC and BDC:

• h ² = b ² – (b × cos α) ²
• h ² = a ² – (c – b × cos α) ²

Equate the right sides of the equations:

• b ² – (b × cos α) ² = a ² – (c – b × cos α) ² 9003 2

or

• a ² = b ² + c ² – 2bc × cos α

If one of the corners at the base is obtuse (the height rests on the continuation of the base), it is completely similar to that considered above.

Define sides b and c:

• b ² = a ² + c ² – 2ac × cos β;
• c ² = a ² + b ² – 2ab × cos γ.

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Statement of the theorem for each side of the triangle

The cosine theorem is valid for all sides of the triangle, i.e.: a

b ² = c ² + a ² – 2ca cos β s γ

The cosine theorem can be used for any kind of triangle .

Cosines of the angles of a triangle

The cosine theorem allows you to find both the cosine and the angle of a triangle. Let’s find the cosines of the angles:

Similarly:

Determination of the angle using the cosine

And now let’s pay attention to the angles.

As we already know, the cosine of an angle from the interval (0°; 180°) determines the angle (as opposed to its sine).

Let us be given a unit semicircle. If we are given cos α, then we are given a point on the upper semicircle and an angle α is given. Therefore, cos α uniquely determines the point M(cos α; sin α), and the angle ∠AOM is uniquely determined.

Consideration of the limits of variation of cos α and sin α

Consider the limits of variation of sine and cosine α. Recall that if α is the angle of a triangle, then it lies between 0° and 180°.

Limit of cosine change: -1 < cos α < 1.

Limit of change of sine: 0 < sin α ≤ 1.

• If cos α > 0, then α ∈ ( 0°;90° )
• If cos α < 0, then α ∈ (90°;180°)
• If cos α = 0, then α = 90°

Examples of solving problems

Using the cosine theorem, you can solve geometry problems. Let’s consider some interesting cases.

Example 1. Triangle ABC is given. Find the length CM.

∠C = 90°, AB = 9, BC = 3, AM/MB = 1/2, where M is a point on the hypotenuse of AB.

How do we decide:

1. Since AM + MB = 9, and AM / MB = 1/2, then AM = 3, MB = 6.
   From the triangle ABC we find cos B:

2. From the triangle CMB by the cosine theorem we find the SM:
   

Answer: CM = .

Example 2. Given a triangle ABC, in which a 2 + b 2 < c 2 . Prove that ∠C is an obtuse angle.

How to prove:

2. Since a ² + b ² < c ² , then cos C < 0, hence ∠C is obtuse.