Cameron Miranda
The law of cosines is a trigonometric rule that relates the lengths of a triangle's sides to the cosine of one of its angles. It can be used to find the length of a triangle's unknown side when the length of the other two sides and the angle between them are known. The law of cosines can be written in various forms, including algebraic notation, symbolic form, and geometric proofs. It can also be generalized to all polyhedra by applying the divergence theorem. The law of cosines has been used since ancient times, with Euclid proving the theorem using Pythagorean theory, and it continues to be an essential tool in solving triangle-related problems today.
| Characteristics | Values |
|---|---|
| Law of Cosines formula | a2 = b2 + c2 – 2bc cos α |
| Finding the unknown angle | cos α = [b2 + c2 – a2]/2bc |
| Finding the second angle | cos β = [a2 + c2 – b2]/2ac |
| Finding the third angle | cos γ = [b2 + a2 – c2]/2ab |
| Relation | The relation between the lengths of sides of a triangle with respect to the cosine of its angle |
| Application | Used to determine the third side of a triangle when the lengths of the other two sides and the angle between them are known |
| Triangle type | Can be used for all types of triangles |
| Proving the law of cosines | By calculating areas, using the Pythagorean theorem, or the geometry of the circle |
| Generalization | Can be generalized to all polyhedra by invoking the divergence theorem |
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What You'll Learn
The law of cosines formula for all triangles
The law of cosines, also known as the cosine rule or cosine formula, is a formula that relates the lengths of a triangle's sides to the cosine of one of its angles. This theorem was first written using algebraic notation by François Viète in the 16th century and was later written in its modern symbolic form in the 19th century.
The law of cosines can be used to find the unknown sides of a triangle when the length of the other two sides and the angle between them are known. The formula for this is:
A^2 = b^2 + c^2 - 2bc x cos(alpha)
Where:
- A is the unknown side
- B and c are the known sides of the triangle
- Alpha is the angle between b and c
Similarly, if beta and gamma are the angles between sides ca and ab, respectively, the formula can be written as:
B^2 = a^2 + c^2 - 2ac x cos(beta)
Or:
C^2 = a^2 + b^2 - 2ab x cos(gamma)
These formulas can be rearranged to find the unknown angles of a triangle when the lengths of all three sides are known. For example, to find angle C, the formula can be rearranged as follows:
C^2 = a^2 + b^2 - 2ab x cos(C)
The law of cosines is a generalisation of the Pythagorean theorem, which only applies to right triangles. The law of cosines can be used for all types of triangles and is particularly useful when solving for triangles with two sides and their included angle given.
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Proving the law of cosines using areas
The Law of Cosines, also known as the Cosine Rule or Cosine Formula, establishes a relationship between the lengths of the sides of a triangle and the cosine of its angle. It can be used to determine the length of the third side of a triangle when the length of the other two sides and the angle between them are known. The law of cosines can be proved using areas.
One way to do this is by considering the areas of specific triangles and then moving on to more general triangles. This can involve cutting polygons into smaller pieces and comparing the areas of the resulting pieces. For example, a hexagon can be cut into smaller pieces in a way that yields a proof of the law of cosines when the angle γ is obtuse. The pieces may include areas a^2, b^2, and -2ab cos γ on one side and c^2 on the other side. By showing that the total areas on both sides are equal, we can prove the law of cosines.
Another way to prove the law of cosines using areas is by applying the Pythagorean theorem to right triangles formed within the original triangle. Euclid's Proposition 12 from Book 2 of the Elements demonstrates this approach. He applies the Pythagorean theorem to two right triangles formed by dropping a perpendicular from one vertex to the side opposite. This approach can be extended to non-right triangles by using the Pythagorean identity, sin^2 θ + cos^2 θ = 1.
The law of cosines can also be proved geometrically by considering the areas of parallelograms formed by the sides of the triangle. If γ is acute, then ab cos γ is the area of the parallelogram with sides a and b forming an angle of γ' = π/2 - γ. If γ is obtuse, then -ab cos γ is the area of the parallelogram with sides a and b forming an angle of γ' = γ.
In summary, the law of cosines can be proved using areas by considering the areas of specific triangles, applying the Pythagorean theorem to right triangles within the original triangle, or by using geometric properties of parallelograms formed by the sides of the triangle. These proofs often involve manipulating the shapes and areas within a triangle to establish the relationship between the side lengths and the cosine of the angle, as described by the law of cosines.
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The obtuse case
The law of cosines, also known as the cosine formula or cosine rule, relates the lengths of the sides of a triangle to the cosine of one of its angles. The theorem was first written using algebraic notation by François Viète in the 16th century.
The law of cosines can be used to find a side in the ambiguous case, where the quadratic formula will tell us how many triangles have the given properties. For example, if the quadratic equation has one positive solution, there is one triangle. If the quadratic equation has two positive solutions, there are two triangles.
The law of cosines can be proven in several ways. One method involves using the Pythagorean theorem, which only holds for right triangles. By applying the Pythagorean theorem to the right triangle on the left-hand side of a figure, we can derive the law of cosines:
C^2 = (b - a*cos(γ))^2 + (a*sin(γ))^2
= b^2 - 2ab*cos(γ) + a^2*cos^2(γ) + a^2*sin^2(γ)
= b^2 + a^2 - 2ab*cos(γ)
This proof requires modification if b < a*cos(γ). In this case, the right triangle used for the Pythagorean theorem is outside the triangle ABC, and the quantity b is replaced by a*cos(γ) - b. This situation arises when β is obtuse, and it can be avoided by reflecting the triangle about the bisector of γ.
Another way to prove the law of cosines is by calculating areas. When the angle γ becomes obtuse, a case distinction is necessary due to the change in sign. If γ is acute, then ab*cos(γ) represents the area of a parallelogram with sides a and b forming an angle of γ' = π/2 - γ. When γ is obtuse, cos(γ) is negative, and -ab*cos(γ) represents the area of the parallelogram with sides a and b forming an angle γ' = γ - π/2. This distinction is illustrated in figures showing a heptagon cut into smaller pieces to prove the law of cosines.
The law of cosines can also be generalized to all polyhedra by considering any polyhedron with vector sides and applying the divergence theorem.
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The acute case
The law of cosines is a formula used to find the unknown side of a triangle when the lengths of the other two sides are given, along with the angle between the two known sides. It can be written in several ways, depending on the values of the sides and angles involved.
When considering the acute case, where angle γ is acute, the law of cosines can be written as:
A2 = b2 + c2 − 2bc cos α
B2 = a2 + c2 − 2ac cos β
C2 = a2 + b2 − 2ab cos γ
In these formulas, a, b, and c represent the sides of the triangle, and α, β, and γ are the angles between the sides. The acute case assumes that angle γ is acute, meaning it is less than 90 degrees.
The law of cosines can also be written in a more concise form, where the focus is on finding the unknown side:
A2 = b2 + c2 − 2bc cos α
Here, a is the unknown side, and b and c are the known sides. This formula can be rearranged to find the other unknown sides by changing the variables accordingly.
Additionally, the law of cosines can be proved by calculating areas. In the acute case, if γ is acute, then ab cos γ represents the area of a parallelogram with sides a and b forming an angle of γ′ = π/2 − γ. This geometric approach provides another way to understand and apply the law of cosines.
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The law of cosines for SSS congruence
The law of cosines is a theorem that was first written using algebraic notation by François Viète in the 16th century. It was later written in its modern symbolic form in the 19th century. The law of cosines can be used to solve SSS triangles, that is, triangles where the three sides are known but the angles are unknown.
To apply the law of cosines to SSS triangles, we can use the formula:
Γ = acos((a² + b² − c²)/(2ab))
This formula can be used to find the angle between sides 'a' and 'b'. To find the angle between sides 'a' and 'c', we can apply the law of cosines again:
Β = acos((a² + c² − b²)/(2ac))
The third angle, α, can be calculated by subtracting the sum of β and γ from 180 degrees:
Α = 180° - β - γ
By applying the law of cosines in this way, we can determine the angles of an SSS triangle.
It is important to note that the law of cosines assumes that the triangle in question is a planar triangle, meaning that the three vertices lie in the same plane. This is because non-planar triangles do not satisfy the theorem.
Additionally, the law of cosines can be used to calculate the area of an SSS triangle. The formula for the area of a triangle with sides a, b, and c is:
Area = ½ ab sin γ
Alternatively, Heron's formula can be used:
Area = √(s(s - a)(s - b)(s - c))
Where s = (a + b + c)/2.
By using these formulas and applying the law of cosines, we can solve SSS triangles and find their angles and areas.
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