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The law of cosines, also known as the cosine rule, is a trigonometric equation that allows us to determine the unknown length of one side of a triangle when the lengths of the other two sides and the size of the included angle are known. It can also be used to calculate the angles of a triangle when the lengths of all three sides are known. The geometric basis of the law of cosines can be attributed to the Greek mathematician Euclid, who laid the groundwork in his work Elements around 300 BCE. However, an explicit statement of the law was first introduced by the Persian mathematician and astronomer Jamshīd al-Kāshī in the 15th century.
| Characteristics | Values |
|---|---|
| Name | Law of Cosines, Cosine Rule, Cosine Law, Al-Kashi Law, Cosine Formula |
| Description | An equation that allows us to find the unknown length of one side of a triangle, provided the lengths of the other two sides and the size of the included angle are known. It also allows us to calculate the angles of a triangle if the lengths of all three sides are known. |
| Formula | c2 = a2 + b^2 - 2ab*cos(γ) |
| Relation to Pythagorean Theorem | A generalization of the Pythagorean theorem that relates the lengths of the sides of any triangle. |
| Use Cases | Finding unknown values in an oblique triangle, triangulation, navigation, geodesy for calculating great-circle distances |
| Contributors | Greek mathematician Euclid, Persian mathematician Jamshīd al-Kāshī |
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What You'll Learn
- The law of cosines is a generalisation of the Pythagorean theorem
- It can be used to find unknown values in an oblique triangle
- The geometric basis of the law was laid by Greek mathematician Euclid
- In the 15th century, Persian mathematician Jamshīd al-Kāshī provided a statement of the law
- The law is also known as Al-Kashi's theorem in France
The law of cosines is a generalisation of the Pythagorean theorem
The law of cosines, sometimes called the cosine rule, is an equation that allows us to find the unknown length of one side of a triangle, given the lengths of the other two sides and the size of the included angle. It can also be used to calculate the angles of a triangle if the lengths of all three sides are known.
The geometric equivalent of the law of cosines can be found in the work of the Greek mathematician Euclid, in his work "The Elements", believed to have been written in the 3rd century BCE. In this work, Euclid essentially extended Pythagoras' theorem to include triangles that were not right-angled triangles.
An explicit statement of the law of cosines first appeared in the work of the Persian mathematician and astronomer Jamshīd al-Kāshī in the 15th century CE. The law of cosines is also known as Al-Kashi's law, in reference to this.
The law of cosines can be stated algebraically as:
C^2 = a^2 + b^2 - 2ab cos(C)
Where a, b, and c are the lengths of the sides of the triangle, and C is the angle opposite side c. This equation can be rearranged to find the length of any side of a triangle, as long as we know the lengths of the other two sides and the included angle, or the size of any angle if we know the lengths of all three sides.
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It can be used to find unknown values in an oblique triangle
The Law of Cosines, also known as the Cosine Rule or Cosine Law, is a trigonometric formula that relates the lengths of a triangle's sides to the cosine of one of its angles. It is a generalisation of the Pythagorean Theorem, applicable to all triangles, not just right triangles.
The Law of Cosines can be used to find unknown values in an oblique triangle. An oblique triangle is a triangle that does not have a 90-degree or "right" angle. To use the Law of Cosines, we need to know the values of either SAS (side-angle-side) or SSS (side-side-side).
The formula for the Law of Cosines is:
> c^2 = a^2 + b^2 - 2ab*cos(γ)
Where a, b, and c are the lengths of the sides of the triangle, and γ is the angle opposite side c.
For example, let's say we have an oblique triangle with sides a = 3, b = 5, and we want to find the length of side c. We also know that the angle opposite side c is 60 degrees. Using the Law of Cosines, we can calculate:
> c^2 = 3^2 + 5^2 - 2(3)(5)cos(60)
> c^2 = 9 + 25 - 30cos(60)
> c^2 = 34 - 15
> c^2 = 19
> c = √19
> c = ~4.36
So, the length of the unknown side c is approximately 4.36 units.
The Law of Cosines can also be used to find the size of missing angles in a triangle, in addition to the length of missing sides. This makes it a powerful tool in trigonometry, providing a general formula that works for all types of triangles.
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The geometric basis of the law was laid by Greek mathematician Euclid
The law of cosines, also known as the cosine rule or cosine formula, is an equation that allows us to find the unknown length of one side of a triangle, given the lengths of the other two sides and the size of the included angle. It can also be used to calculate the angles of a triangle if the lengths of all three sides are known.
The geometric basis of the law of cosines was laid by the Greek mathematician Euclid in his work "The Elements", believed to have been written during the third century BCE. Euclid's work extended Pythagoras' theorem beyond right-angled triangles to include triangles that were not right-angled, such as oblique triangles.
Euclid's Proposition 12 from Book II of "The Elements" states that in an oblique triangle containing an obtuse angle, the square of the longest side (the side opposite the obtuse angle) is greater than the sum of the squares of the other two sides. To solve this, Euclid extended one of the shorter sides to create a right-angled triangle. By doubling the area of a rectangle equal to the product of the sides of the triangle, he was able to determine the difference in area.
Euclid's work on the geometric equivalent of the law of cosines was further developed by later mathematicians. For example, in the 11th century, al-Bīrūnī used Euclid's work to solve triangles in the context of astronomical problems. In the 13th century, the Persian mathematician Naṣīr al-Dīn al-Ṭūsī described how to solve triangles from various combinations of given data in his work "Kitāb al-Shakl al-qattāʴ" (Book on the Complete Quadrilateral).
The law of cosines was first written using algebraic notation by François Viète in the 16th century. The modern algebraic notation we use today was developed at the beginning of the 19th century.
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In the 15th century, Persian mathematician Jamshīd al-Kāshī provided a statement of the law
In the 15th century, the Persian mathematician Jamshīd al-Kāshī provided a statement of the law of cosines, also known as the cosine rule or cosine formula. Al-Kāshī's statement built upon the work of earlier mathematicians, including the Greek Euclid, who is known for his work on Pythagoras' theorem.
Al-Kāshī's work, "Miftāḥ al-ḥisāb" ("Key of Arithmetic", 1427), included a description of the solution of triangles from various combinations of given data. In particular, he described a method for finding the unknown length of one side of a triangle when the lengths of the other two sides and the size of the included angle are known. This is known as the law of cosines or the cosine rule, which can be stated algebraically as:
$$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$$
Where $a$, $b$, and $c$ are the lengths of the sides of the triangle, and $C$ is the angle opposite side $c$. This formula allows us to calculate the length of the unknown side $c$ by knowing the lengths of the other two sides and the included angle.
Al-Kāshī's statement of the law of cosines was a significant contribution to trigonometry and built upon the work of earlier mathematicians. It provided a concise and consolidated formula for solving triangles and finding unknown side lengths or angles, making it a valuable tool in mathematics and geometry.
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The law is also known as Al-Kashi's theorem in France
The Law of Cosines, also known as the Cosine Rule or Cosine Law, is a trigonometric statement about general triangles. It relates the lengths of a triangle's sides to the cosine of one of its angles. Using the conventional triangle notation, the law of cosines states that:
> c^2 = a^2 + b^2 − 2ab cos(γ), or, equivalently: b^2 = c^2 + a^2 − 2ca cos(β), a^2 = b^2 + c^2 --
Where a, b, and c are the lengths of the sides of the triangle, and C is the angle opposite side c. The law of cosines is a generalization of the Pythagorean theorem, which only holds for right triangles.
The law of cosines was first formulated by the Persian mathematician and astronomer Jamshīd al-Kāshī in the fifteenth century CE. In France, the law is known as Al-Kashi's theorem. Al-Kashi's formulation built on the work of Greek mathematician Euclid, who, in the third century BCE, extended Pythagoras' theorem to include triangles that were not right-angled. The geometric equivalent of the law of cosines can also be found in Euclid's work.
The law of cosines has several applications. It can be used to find the size of missing angles, as well as the length of missing sides. It can also be generalized to all polyhedra by considering any polyhedron with vector sides and invoking the divergence theorem.
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Frequently asked questions
The law of cosines, also known as the cosine rule or cosine formula, was first formulated by the Persian mathematician and astronomer Jamshīd al-Kāshī in the fifteenth century CE.
The law of cosines is used to find the length of one side of a triangle when the lengths of the other two sides and the size of the included angle are known. It can also be used to calculate the angles of a triangle if the lengths of all three sides are known.
In French, the law of cosines is referred to as "Le théorème d’Al-Kashi", which translates to "Al-Kashi's Theorem".
The law of cosines is a generalisation of the Pythagorean theorem, relating the lengths of the sides of any triangle, not just right-angled triangles.
