Gavin Howe
The law of cosines, also known as the cosine rule, is a formula that can be used to determine the unknown side of a triangle when the length of the other two sides and the angle between them are known. It can be used for all types of triangles, including right triangles, and is particularly useful when we need to find unknown values in an oblique triangle. The law of cosines generalizes the Pythagorean theorem, which only holds for right triangles.
| Characteristics | Values |
|---|---|
| Name | Law of Cosines |
| Other Names | Cosine Rule, Cosine Formula |
| Formula | a² = b² + c² – 2bc cos α |
| Formula Variables | a, b, and c are the sides of a triangle and α is the angle between sides b and c |
| Application | Used to find the unknown side of a triangle when the length of the other two sides are given and the angle between the two known sides |
| Application | Can be used for all types of triangles |
| Application | Can be used to find unknown angles |
| Application | Can be used to find missing angles in SSS congruence |
| Application | Can be used to find unknown values in an oblique triangle |
| Application | Can be used when the values of SAS (side-angle-side) are known |
| History | First written using algebraic notation by François Viète in the 16th century |
| History | Modern algebraic notation allowed the law of cosines to be written in its current form in the 19th century |
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What You'll Learn
- The Law of Cosines is a general formula that works for all types of triangles
- The Law of Cosines is used to determine the third side of a triangle
- The Law of Cosines is also called the Cosine Rule
- The Law of Cosines can be used to find unknown values in an oblique triangle
- The Law of Cosines can be used with the Law of Sines to find all values
The Law of Cosines is a general formula that works for all types of triangles
The Law of Cosines, also known as the Cosine Rule or Cosine Formula, is a general formula that works for all types of triangles. It is used to determine the unknown side of a triangle when the length of the other two sides and the angle between them are known. This can be represented by the formula:
A^2 = b^2 + c^2 - 2bc cos(α)
Where a, b, and c are the sides of the triangle, and α is the angle between sides b and c. This formula can be rearranged to solve for different known values. For example, if we know the values of two sides and one angle (SAS), we can use the Law of Cosines to find the unknown side.
The Law of Cosines is particularly useful for oblique triangles, which lack a 90-degree or "right" angle. In the case of right triangles, the Law of Cosines simplifies to the Pythagorean theorem, as the cosine of a right angle is 0. However, the Law of Cosines is not restricted to right triangles and can be applied to all types of triangles, including acute and obtuse triangles.
The theorem was first written using algebraic notation by François Viète in the 16th century, and Euclid later proved it by applying the Pythagorean theorem to right triangles. The Law of Cosines is a valuable tool for solving triangles and can be used alongside the Law of Sines to find unknown sides and angles.
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The Law of Cosines is used to determine the third side of a triangle
The Law of Cosines is a general formula that can be used to determine the unknown side of a triangle. It is applicable to all types of triangles, including right triangles, and does not rely on creating an "imaginary" right triangle. Instead, it generalizes the relationship between side lengths and angles in any triangle.
The Law of Cosines is particularly useful when we need to find unknown values in an oblique triangle, or a triangle that does not have a 90-degree angle. In such cases, we can use the Law of Cosines if we know the values of SAS (side-angle-side) or SSS (side-side-side).
The formula for the Law of Cosines is:
> a^2 = b^2 + c^2 - 2bc cos α
Where a, b, and c are the sides of the triangle, and α is the angle between sides b and c. By using this formula, we can find the length of the third side of a triangle when we know the lengths of the other two sides and the angle between them.
For example, let's consider a triangle ABC with sides a = 10cm, b = 7cm, and c = 5cm. We can use the Law of Cosines to find the measure of the unknown angle. First, we find the value of one angle using the formula:
> cos α = (b^2 + c^2 - a^2) / (2bc)
Then, we can find the second angle using the same law:
> cos β = (a^2 + c^2 - b^2) / (2ac)
Finally, we can determine the third angle using the angle sum property of triangles, which states that the sum of all three angles in a triangle is equal to 180 degrees.
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The Law of Cosines is also called the Cosine Rule
The Law of Cosines, also known as the Cosine Rule or Cosine Formula, is a trigonometric principle that relates the lengths of a triangle's sides to the cosine of one of its angles. In other words, it helps determine the length of the unknown sides of a triangle when the lengths of the other two sides and the angle between them are known. This law is expressed by the formula:
> c^2 = a^2 + b^2 - 2ab cos(C)
Where a, b, and c represent the sides of a triangle, and C is the angle between sides a and b. This formula can be rearranged to find the other angles as well.
The Law of Cosines is a generalization of the Pythagorean theorem, which only applies to right triangles. The Pythagorean theorem states that for a right triangle with sides a, b, and c, where a is the hypotenuse, the following equation holds true:
> a^2 + b^2 = c^2
However, the Law of Cosines is not restricted to right triangles and can be applied to all types of triangles, including acute and obtuse triangles. It is useful when solving triangles with all three sides or two sides and their included angle given.
The Law of Cosines was first written using algebraic notation by François Viète in the 16th century, and Euclid later proved it by applying the Pythagorean theorem to each of the two right triangles.
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The Law of Cosines can be used to find unknown values in an oblique triangle
The Law of Cosines, also known as the Cosine Rule or Cosine Formula, is a powerful tool in trigonometry. It relates the lengths of a triangle's sides to the cosines of its angles. This law is not restricted to right triangles; it can be applied to all triangles, including oblique triangles, to find unknown values.
The Law of Cosines is particularly useful when we need to find unknown values in an oblique triangle. An oblique triangle is a triangle that lacks a 90-degree or "right" angle. In such cases, we can use the Law of Cosines if we know the values of SAS (side-angle-side) or SSS (side-side-side). For example, if we know the length of two sides and the angle between them, we can use the Law of Cosines to determine the length of the third side.
The formula for the Law of Cosines is:
> a^2 = b^2 + c^2 - 2bc cos(α)
Where a, b, and c are the sides of the triangle, and α is the angle between sides b and c.
Additionally, the Law of Cosines can be used to find unknown angles in a triangle. The formula to find the unknown angles is:
> cos(α) = (b^2 + c^2 - a^2) / (2bc)
This formula allows us to calculate the cosine of angle α using the lengths of the sides of the triangle.
It's important to note that the Law of Cosines is a generalization of the Pythagorean theorem, which only holds for right triangles. When applied to a right triangle, the Law of Cosines reduces to the Pythagorean theorem, as the cosine of a right angle is 0. However, the Law of Cosines provides a more flexible approach that can be used for all types of triangles, making it a valuable tool in trigonometry and mathematics.
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The Law of Cosines can be used with the Law of Sines to find all values
A^2 = b^2 + c^2 - 2bc cos α
Where a, b, and c are the sides of the triangle, and α is the angle between sides b and c.
The Law of Cosines can also be used to find the unknown angles of a triangle when all three sides are known. The formula for this is:
Cos α = (b^2 + c^2 - a^2) / (2bc)
Cos β = (a^2 + c^2 - b^2) / (2ac)
Cos γ = (b^2 + a^2 - c^2) / (2ab)
The Law of Sines, also known as the Sine Rule or Sine Formula, is used to find the unknown angle or unknown side of a triangle. It is defined as the ratio of the sides of a triangle to the sine of their opposite angle. The formula for this is:
A/sin A = b/sin B = c/sin C
The Law of Sines can be used to find the unknown side of a triangle when two angles and one side are known, or when two sides and one non-included angle are given. It can also be used to find the unknown angle of a triangle when two sides and one included angle are known.
Both the Law of Cosines and the Law of Sines can be used together to find all the values in a triangle. For example, if we know the length of two sides and the angle between them, we can use the Law of Cosines to find the length of the third side. Then, we can use the Law of Sines to find the other two angles of the triangle. Alternatively, if we know the length of all three sides of a triangle, we can use the Law of Sines to find all the angles.
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Frequently asked questions
Yes, the law of cosines can be applied to right triangles. It is a more general formula that works for all types of triangles, not just those with a right angle.
The law of cosines, also known as the cosine rule, is used to determine the third side of a triangle when the lengths of the other two sides and the angle between them are known.
The law of cosines applied to right triangles is the Pythagorean theorem, since the cosine of a right angle is 0. For example, if sides a and b and the angle between them are known, then the length of the third side, c, can be found using the formula: a^2 + b^2 - 2ab.c = c^2.
The law of sines can be used in any triangle (not just right triangles) where a side and its opposite angle are known. The law of cosines, on the other hand, can be used in any triangle where you are trying to relate all three sides to one angle.
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