To find any side of a triangle, the law of cosine is transformed when one knows two lengths of side and the dimensions of an angle opposite to one of them. See how our cosine calculator does all the math for you! For a right triangle, the gamma angle, i.e. the angle between legs a and b, is equal to 90°. The cosine of 90° = 0, so in this particular case the formula of the law of cosine is reduced to the well-known equation of the Pythagorean theorem: there are several ways to prove the law of the cosine equation. You`ve already read about one of them – it comes directly from Euclid`s formulation of the law and an application of Pythagoras` theorem. You can write the other proofs of the law of cosine with: You can also combine the calculator for the law of cosine with the law of sine to solve other problems, such as: to find the side of the triangle if two of the angles and one side (AAS and ASA) are given. The law of cosine states that for a triangle with sides and angles denoted by symbols, as noted above, the law of cosine is a set of formulas that relate the length of the sides of a triangle to one of its cosine angles. The law of cosine is usually preferred when three sides of a triangle are given to find any angle A, B or C of the triangle or the two adjacent sides, and an angle is given. Use a free online cosine calculator that helps find unknown lengths of sides or angles of a triangle. You can calculate all the other remaining sides and measure the angles in a triangle using different forms of the cosine law.

You can transform these cosine formulas to solve some triangulation problems (solving a triangle). You can use them to find: The last two proofs require distinguishing different triangular cases. The one based on the definition of the point product is shown in another article, and the proof with the law of the sine is quite complicated, so we decided not to reproduce it here. If you are curious about this cosine proof, read Wikipedia`s explanation. The law of cosine is a generalization of the Pythagorean theorem, so that whenever the latter works, the former can also be applied. But not the other way around! To complete the proof of the law of cosine, you need to add equations (1) and (2) and subtract (3): The cosine law calculator can help you solve a variety of triangular problems. You will learn about the law of cosine (also known as the rule of cosine), the law of cosine and its applications. Scroll down to find out when and how you can apply the law of cosine and see the evidence for that law. With this triangle calculator, you can quickly find the properties of any triangle.

The law of cosine is one of the fundamental laws and is often used for many geometric problems. We also use this law in many omnitools, to name a few: The calculator for the law of cosine is 100% free to find the sides and angles of a triangle. Manually calculating all trigonometry measurements is a very complex task that increases the probability of errors. With this handy calculator, you can avoid the risk of getting accurate values, and this online calculator helps students and tutors solve the law of cosine. The law of the cosine theorem is a form of the Pythagorean theorem adapted to use the non-rectangular triangle, but the Pythagorean theorem only works for right triangles. Thus, you can use the cosine calculator to calculate any value of sides and angles of a triangle. Enter the known values in the corresponding fields of this triangle calculator. Don`t forget to use the image above to verify that you have marked the sides and angles with the correct symbols. If you want to save time, enter lateral lengths into our law of sinusoidal calculators – our tool is a safe bet! Just follow these simple steps: If the two lateral lengths and angle are known, just paste the values into the cosine calculator and let it perform calculations. Another relatively easy-to-understand law of cosine uses Ptolemy`s theorem: if you know the three sides of a triangle, you can use the cosine rule to find the angles of a triangle. Thus, the solution formula for angles used by the cosine law formula is: The cosine law (alternatively the cosine formula or the cosine rule) describes the relationship between the lengths of the sides of a triangle and the cosine of its angles.

It can be applied to all triangles, not just right-angled triangles. This law generalizes the Pythagorean theorem because it allows to calculate the length of one of the sides, provided you know the length of the other two sides and the angle between them. But if you`re somehow wondering what cosine is, you`d better take a look at our cosine calculator. No, the law of cosine applies to all triangles. In fact, if you apply the law of cosine to a right triangle, you will arrive at the good old Pythagorean theorem. Let`s say we drew the triangle ABC in its perimeter, as in the image. The law of cosine is a modified version of the Pythagorean theorem used to find unknown values of the sides and angles of non-rectangular triangles. To embed this widget in a publication, install tungsten| Alpha Widget Shortcode plugin and copy and paste the above shortcode into the HTML source code.

α = arccos [(b² + c² – a²)/(2bc)] β = arccos [(a² + c² – b²)/(2ac)] γ = arccos [(a² + b² – c²)/(2ab)] Similar equations can be derived for two other sides:. From Wikipedia source: Ptolemy`s theorem, blunt case, Pythagoras` theorem, distance formula. If the length of sides a, b and c is opposite to angles A, B and C. Then the law of cosine expresses:. Zone A = (sqrt {s*(s – a)*(s – b)*(s – c)}) Select the parameter you want to calculate and enter the required parameters in the appropriate fields. The tool uses the law of cosine to determine results instantly.