The law of cosines calculator can help you solve problems that include triangles. You will learn what is the law of cosines (sometimes it's called the cosine rule), the law of cosines formula and its applications. Scroll down to find out when and how to use the law of cosines and check out the proofs of this law. Thanks to this triangle calculator, you will be able to find the properties of any arbitrary triangle quickly.

triangle sides and angles, law of cosines formula

Law of cosines formula

The law of cosines states that, for a triangle with sides and angles denoted with symbols as in the picture above,

a² = b² + c² - 2bc * cos(α)

b² = a² + c² - 2ac * cos(β)

c² = a² + b² - 2ab * cos(γ)

For a right triangle, the angle gamma, which is the angle between legs a and b, is equal to 90°. The cosine of 90° = 0, so in that special case, the law of cosines formula is reduced to the well-known equation of Pythagorean theorem:

a² = b² + c² - 2bc * cos(90°)

a² = b² + c²

What is the law of cosines

The law of cosines (also known as the cosine formula or cosine rule) describes the relationship between the lengths of the sides and the cosine of one of its angles. It can be applied to all triangles, not only the right triangles. This law generalizes the Pythagorean theorem, as it allows to calculate the length of one of the sides, knowing the other sides and the angle between them.

The law appeared in Euclid's Element, a mathematical treatise containing definitions, postulates, and geometry theorems. It wasn't formulated in the way we learn it now, as the concept of cosine was not developed yet.

triangle sides and angles, image for Euclid's proof

AB² = CA² + CB² - 2 * CA * CH (for acute angles, '+' for obtuse)

However, the Euclid's theorem may be reformulated easily to the current cosine formula form:

CH = CB * cos(γ), so AB² = CA² + CB² - 2 * CA * (CB * cos(γ))

Changing notation, we obtain the familiar expression:

c² = a² + b² - 2ab * cos(γ)

The first explicit equation of cosine rule was presented by Persian mathematician d'Al-Kashi in the 15th century, in the 16th century the law was popularized by famous French mathematician Viète, to receive the final shape in the 19th century.

When to use the law of cosines - applications

You can transform these law of cosines formulas to solve some problems of triangulation (solving a triangle).You can use them to find:

  1. The third side of a triangle, knowing two sides and the angle between them (SAS):
  • a = √[b² + c² - 2bc * cos(α)]
  • b = √[a² + c² - 2ac * cos(β)]
  • c = √[a² + b² - 2ab * cos(γ)]
  1. The angles of a triangle, knowing all three sides (SSS):
  • α = arccos [(b² + c² - a²)/(2bc)]
  • β = arccos [(a² + c² - b²)/(2ac)]
  • γ = arccos [(a² + b² - c²)/(2ab)]
  1. The third side of a triangle, knowing two sides and an angle opposite to one of them (SSA):
  • a = b*cos(γ) ± √[c² - b²*sin²(γ)]
  • b = c*cos(α) ± √[a² - c²*sin²(α)]
  • c = a*cos(β) ± √[b² - a²*sin²(β)]

Just remember that knowing two sides and an adjacent angle can yield two distinct possible triangles (or one or zero positive solutions, depending on the given data). That's why we've decided to implement SAS and SSS in this tool, but not SSA.

Law of cosines is one of the basic laws and it's widely used for many geometric problems. We also take advantage of that law in many omnitools, to mention only a few:

Also, you can combine the law of cosines calculator with the law of sines to solve other problems, for example finding the side of the triangle, given two of the angles and one side (AAS and ASA).

Law of cosines proofs

There are many ways in which the law of cosines equation may be proved. You've already read about one of them - it comes directly from Euclid's formulation of the law and from an application of the Pythagorean theorem. The other proofs of the law of cosines can be written using:

  1. Trigonometry
triangle sides and angles, image for trigonometric proof

Altitude of a triangle divide the opposite side into two parts: b = b₁ + b₂ From sine and cosine definitions, b₁ might be expressed as a * cos(γ) and b₂ = c * cos(α). Hence:

b = a * cos(γ) + c * cos(α) and by multiplying it by b, we get:

b² = ab * cos(γ) + bc * cos(α) (1)

Analogical equations may be derived for other two sides:

a² = ac * cos(β) + ab * cos(γ) (2)

c² = bc * cos(α) + ac * cos(β) (3)

To finish the law of cosines proof, you need to add the equation (1) and (2) and subtract (3):

a² + b² - c² = ac * cos(β) + ab * cos(γ) + bc * cos(α) + ab * cos(γ) - bc * cos(α) - ac * cos(β)

Reduction and simplification of the equation give one of the forms of the cosine rule:

a² + b² - c² = 2ab * cos(γ)

c² = a² + b² - 2ab * cos(γ)

  1. Distance formula
triangle sides and angles in Cartesian coordinate system, image for proof using distance formula

Let C = (0,0), A = (b,0), as in the image.

To find the coordinates of B, we can use the definition of sine and cosine:

B = (a * cos(γ), a * sin(γ))

From the distance formula, we can find that

c = √[(x₂ - x₁)² + (y₂ - y₁)²] = √[(a * cos(γ) - b)² + (a * sin(γ) - 0)²]

Thus

c² = a² * cos(γ)² - 2ab * cos(γ) + b² + a² * sin(γ)²

c² = b² + a²(sin(γ)² + cos(γ)²) - 2ab * cos(γ)

As a sum of squares of sine and cosine is equal to 1, we obtain the final formula:

c² = a² + b² - 2ab * cos(γ)

  1. Ptolemy's theorem
circumcircle of a triangle with triangle sides and angles, image for proof using Ptolemy's theorem Another law of cosines proof quite easy to understand is the evidence using Ptolemy's theorem:
  • Assume we have the triangle ABC drawn in its circumcircle, as in the picture.

  • We construct congruent triangle ADC, where AD = BC and DC = BA

  • The altitudes from point D and B split the base AC. CE equals FA.

  • From cosine definition we can express CE as a * cos(γ), .

  • Thus, we can write that BD = EF = AC - 2 * CE = b - 2 * a * cos(γ)

  • Then for our quadrilateral ADBC we can use the Ptolemy's theorem, which explains the relation between the four sides and two diagonals. The theorem states that for cyclic quadrilaterals the sum of products of opposite sides is equal to the product of the two diagonals:

    BC * DA + CA * BD = AB * CD

    so in our case:

    a² + b * (b - 2 * a * cos(γ)) + a² = c²

  • After reduction we get the final formula:

    c² = a² + b² - 2ab * cos(γ))

The great advantage of these three proofs is their universality - they work for acute, right and obtuse triangles.

  1. Using the law of sines
  2. Comparison of areas
  3. Geometry of the circle

The last two proofs require the distinction between different triangle cases and the proof using the law of sines is quite complicated, so we don't show them here. If you're curious about these law of cosines proofs, check out the Wikipedia explanation.

How to use the law of cosines calculator

  1. Start with formulating your problem. For example, you may know two sides of the triangle and angle between them and look for the remaining side.

  2. Input the known values into the appropriate boxes of this triangle calculator. Remember to double-check with the figure above whether you denoted the sides and angles with correct symbols.

  3. Watch our law of cosines calculator performing all calculations for you!

Law of cosines - SSS example

If your task is to find the angles of a triangle given all three triangle sides, all you need to do is to use the transformed cosine rule formulas:

  • α = arccos [(b² + c² - a²)/(2bc)]
  • β = arccos [(a² + c² - b²)/(2ac)]
  • γ = arccos [(a² + b² - c²)/(2ab)]

Let's calculate one of the angles. Assume we have a = 4 in, b = 5 in and c = 6 in. We'll use the first equation to find α:

α = arccos [(b² + c² - a²)/(2bc)]

= arccos [(5² + 6² - 4²)/(2 * 5 * 6)]

= arccos [(25 + 36 - 16)/60]

= arccos [(45/60)] = arccos [0.75]

α = 41.41°

The second angle may be calculated from the second equation in an analogical way, and the third angle you can find knowing that sum of the angles in a triangle is equal to 180° (π/2).

If you want to save some time, type the side lengths into our law of sines calculator - also in this case our tool is a safe bet! Just follow these simple steps:

  1. Choose the option depending on given values. We need to pick the second option - SSS (3 sides).

  2. Enter the known values. Type the sides: a = 4 in, b = 5 in and c = 6 in.

  3. The calculator displays the result! In our case the angles are equal to α = 41.41°, β = 55.77° and γ = 82.82°.

After such explanation, we're sure that you understand what the law of cosine is and when to use it. Give this tool a try, solve some exercises and remember that practice makes permanent!

Bogna Haponiuk and Hanna Pamuła, PhD student

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