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.

Understanding Sinus Pain and Congestion: An Overview

Sinus pain and congestion are common conditions that affect many individuals, particularly during cold and allergy seasons. These symptoms can be uncomfortable and sometimes debilitating, impacting daily life and overall well-being. To effectively manage and treat sinus-related issues, it’s crucial to understand their causes, symptoms, and available treatment options.

Common Symptoms of Sinus Pain and Congestion

Recognizing the symptoms of sinus pain and congestion is the first step towards proper management and treatment. Here are some of the most common signs to look out for:

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

Is sinus pain common in young children. Interestingly, sinus pain is not a typical symptom in children under 5 years of age. This is due to the fact that sinuses are not fully developed in young children, making sinus-related issues less common in this age group.

Identifying the Causes of Sinus Congestion

Understanding the underlying causes of sinus congestion is essential for proper treatment. There are three primary causes of sinus congestion:

1. Viral Sinus Infection

Viral sinus infections are the most common cause of sinus congestion. They are typically associated with the common cold and affect the lining of the nose and all the sinuses. These infections are self-limiting and usually resolve on their own within a week or two.

2. Bacterial Sinus Infection

Bacterial sinus infections occur in about 5% of cold cases. They often start as viral infections but become complicated by bacterial overgrowth. Symptoms of bacterial sinus infections include increased sinus pain, return of fever, and possible redness or swelling around the eyelids or cheeks.

3. Allergic Sinus Reaction

Allergic sinus reactions are common in individuals with nasal allergies, such as those triggered by pollen. Symptoms often include sneezing, an itchy nose, and clear nasal discharge in addition to sinus congestion.

The Progression of Nasal Discharge in Sinus Infections

The color and consistency of nasal discharge can provide valuable insights into the progression of a sinus infection. Contrary to popular belief, the color of nasal discharge alone is not a reliable indicator of whether an infection is viral or bacterial.

How does nasal discharge change during a sinus infection. The typical progression is as follows:

1. Clear discharge at the onset of infection
2. Cloudy discharge as the infection progresses
3. Yellow or green-colored discharge for a few days, which is still considered normal

It’s important to note that colored discharge is common after sleep, when using allergy medications, or in low humidity environments. This is due to a decrease in the amount of normal nasal secretions under these conditions.

Recognizing Signs of a Bacterial Sinus Infection

While both viral and bacterial sinus infections can produce yellow or green nasal discharge, there are specific symptoms that may indicate a bacterial infection:

• Thick, pus-like nasal discharge
• Sinus pain (not just normal congestion)
• Swelling or redness of the skin over any sinus
• Fever lasting more than 3 days
• Fever returning after being gone for over 24 hours
• Nasal discharge and post-nasal drip lasting over 14 days without improvement

Can bacterial sinus infections be diagnosed solely based on symptoms. While these symptoms can strongly suggest a bacterial infection, a definitive diagnosis often requires a medical examination and possibly additional tests.

Treatment Options for Sinus Congestion

The appropriate treatment for sinus congestion depends on its underlying cause. Here are the general treatment approaches for different types of sinus congestion:

Viral Sinus Infection Treatment

For viral sinus infections, the primary treatment involves managing symptoms and supporting the body’s natural healing process. Key treatments include:

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

Are antibiotics effective for viral sinus infections. No, antibiotics are not helpful for viral infections and should be avoided to prevent antibiotic resistance.

Bacterial Sinus Infection Treatment

Bacterial sinus infections typically require antibiotic treatment. The specific antibiotic and duration of treatment will be determined by a healthcare provider based on the severity of the infection and the patient’s medical history.

Allergic Sinus Reaction Treatment

For allergic sinus reactions, treatment focuses on managing the underlying allergy. This may include:

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

How do allergy treatments help with sinus symptoms. By reducing the allergic response, these treatments often alleviate associated sinus congestion and inflammation.

Home Remedies for Sinus Congestion

In addition to medical treatments, several home remedies can help alleviate sinus congestion and promote healing:

• Nasal saline rinses to clear thick secretions
• Steam inhalation to moisturize nasal passages
• Elevated head position during sleep to promote sinus drainage
• Warm compresses applied to the face to relieve pain and pressure
• Staying hydrated to thin mucus secretions
• Using a humidifier to add moisture to the air

Are nasal saline rinses effective for all types of sinus congestion. Yes, nasal saline rinses can be beneficial for viral, bacterial, and allergic sinus conditions by helping to clear nasal passages and reduce inflammation.

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. These include:

• Severe trouble breathing
• Redness or swelling on the cheek, forehead, or around the eye
• Severe headache that is worsening
• Fever over 104° F (40° C)
• Symptoms in individuals with weakened immune systems
• Sinus pain and fever that don’t improve with home treatment
• Symptoms lasting more than 10-14 days without improvement

Should you seek medical attention for every case of sinus congestion. No, most cases of sinus congestion, especially those associated with common colds, can be managed at home. However, if symptoms are severe, persistent, or accompanied by the signs mentioned above, it’s important to consult a healthcare provider.

Understanding sinus pain and congestion, their causes, and appropriate treatment options can help individuals manage these common conditions more effectively. By recognizing the signs of more serious infections and knowing when to seek medical attention, people can ensure they receive proper care and avoid complications. Remember, while home remedies and over-the-counter treatments can provide relief in many cases, persistent or severe symptoms should always be evaluated by a healthcare professional.

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.

• 
  
  
  
  
  Bellevue
  
  
  
• 
  
  
  
  
  Everett
  
  
  
• 
  
  
  
  
  Federal Way
  
  
  
• 
  
  
  
  
  Seattle
  
  
  
• 
  
  
  
  
  Virtual Urgent Care
  
  
  

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

309976

4

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

Other stories

Are allergies bad in your state?

Next Story

Spring allergies: Do you have a plan?

Previous Story

Sine theorem. Formulas and Proofs

Let’s help you understand and love math

Start learning

159.4K

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.

Find out what professions of the future will suit you

Take the test and we will show you who you can become, and we will also send you a detailed guide on how to realize yourself right now

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.

For those who want to connect their lives with the exact sciences, Skysmart offers courses in specialized mathematics.

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.

Math cheat sheets for parents

All math formulas at hand

Lidia Kazantseva

By Skysmart

To the previous article

386.7K

9 0235 Places and classes of numbers

To the next article

235.1K

Direct and inverse proportionality

Get a learning plan that will help you understand and love math

At an introductory lesson with a tutor learning

The theorem of cosines and sines of a triangle.

Formulas and examples

We will help you understand and love mathematics

Start learning

262.2K

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.

Tip: to get to grips with a complex topic faster, sign up for
online courses in mathematics for children and teenagers.

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.

Solve your math homework for 5.

Detailed solutions will help you understand the most difficult topic.

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.