Cameron Miranda
The Law of Sines and the Law of Cosines are trigonometric rules that can be used to solve for unknown sides and angles in a triangle. The Law of Sines (or Sine Rule) can be used for any triangle when you know a side and its opposite angle and another side. The Law of Cosines (or Cosine Rule) is more general and can be used for all types of triangles, including oblique triangles, which lack a right angle. It is used when you know the values of two sides and one non-included angle (SAS) or the lengths of all three sides (SSS).
| Characteristics | Values |
|---|---|
| Law of Sines | Used when you have 2 sides and the non-included angle or 2 angles and the non-included side |
| Law of Cosines | Used when you know the lengths of 3 sides and want to find an angle or when you have 2 sides, their included angle, and want to find the side length opposite the angle |
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What You'll Learn
When you have two sides and one angle, but none of the sides are opposite the angle
The Law of Sines and the Law of Cosines are trigonometric formulas used to solve for angles and sides in triangles. The Law of Sines is used when you know two angles and their opposite sides and need to find the third angle or side. It is also used when you have two sides and one non-included angle, and you want to find the unknown side or angle.
The Law of Cosines, on the other hand, is used when you know three sides and one angle, or two sides and one angle (SAS, SSS, or ASA). It is generally preferred over the Law of Sines as it does not introduce ambiguous cases and extraneous solutions. When you have two sides and one angle, but none of the sides are opposite the angle, you have to use the Law of Cosines. This is because the Law of Sines relates two sides and their two opposite angles.
For example, if you know the length of one side and the angle opposite it, as well as the size of the angle adjacent to that side, you can use the Law of Sines to find the length of the side opposite the known angle. However, if none of the sides are opposite the given angle, you would use the Law of Cosines.
To summarise, when you have two sides and one non-adjacent angle, you can use the Law of Sines to solve for the unknown angle or side. However, if none of the sides are opposite the given angle, you must use the Law of Cosines.
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When you have three sides and one angle
The law of sines and the law of cosines are both used to find the unknown sides or angles of a triangle. The law of sines, also known as the sine rule or sine formula, states that the ratio of the side length of a triangle to the sine of the opposite angle is the same for all three sides. In other words, for sides "a", "b", and "c" and angles "A", "B", and "C", the equation (a/sin A) = (b/sin B) = (c/sin C) can be used to find unknown sides or angles. The law of sines is used when two angles and one side, or two angles and one included side, are known. This means that the law of sines can be used when we have ASA (angle-side-angle) or AAS (angle-angle-side) criteria.
The law of cosines, on the other hand, can be used to find an unknown side or angle when the three sides of a triangle are known. For example, if we want to find angle "A" given sides "a", "b", and "c", we can use the law of cosines. However, it is important to note that when using the law of cosines, we should start by finding the angle opposite the longest side to avoid ambiguity, especially if we plan to use the law of sines subsequently. This is because, in some cases, finding the angle opposite the shortest side first can create multiple solutions or no solution at all.
For instance, let's consider a triangle with sides "a" = 1.82, "b" = 2.31, and "c" = 3.30. If we use the law of cosines to find angle "A" first, we get 32.11°. However, if we use the law of cosines to find angle "C" first, we get 105.46°. If we had used the law of sines to find angle "C" after finding angle "A" first, we would have obtained a different value of 74.56°. Therefore, it is recommended to find the angle opposite the longest side first when using the law of cosines, especially if we intend to apply the law of sines subsequently.
In summary, when we have three sides and one angle of a triangle, we can use the law of cosines to find the unknown angle or side. The law of cosines applies to any side of any triangle and can help us determine the necessary values when three sides are given. However, it is important to be mindful of the order in which we find the angles, especially if we plan to use the law of sines afterward, as starting with the angle opposite the longest side helps avoid ambiguity and multiple solutions.
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When you have two angles and one side
The law of sines, also known as the sine rule, can be used to find the unknown side of a triangle when given two angles and one side. This is also known as the ASA (angle-side-angle) or AAS (angle-angle-side) criteria.
The law of sines is defined as the ratio of side length to the sine of the opposite angle. In other words, the side 'a' divided by the sine of angle A is equal to side 'b' divided by the sine of angle B, which is also equal to side 'c' divided by the sine of angle C.
For example, let's say we have a triangle with sides AB = c, BC = a, and AC = b. We can use the sine rule to find the unknown length of one of these sides if we know the lengths of the other two sides and their respective angles.
The law of cosines can also be used to find an unknown angle in a triangle, but the law of sines is generally considered easier due to having fewer operations to perform.
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When you want to find the length of sides of non-right triangles
The Law of Sines and the Law of Cosines are used to determine the length of sides or measure of angles of triangles that are not right triangles.
The Law of Sines is used when you have two sides of a triangle and the angle opposite one of them, or when you have two angles and the side between them. It's about opposite pairs. For example, if you have a side of length 16 opposite a known angle of 115° and you want to find the angle opposite the known side of length 32, you would use the Law of Sines.
The Law of Cosines is used when you know the lengths of two sides and the angle between them, or when you know the lengths of all three sides and want to find the measure of one of the angles. For example, if you know the lengths of sides a and b, and you also know the measurement of the angle between sides a and b, you would use the Law of Cosines.
When solving a problem, look at the angles and side lengths given, and what side length or angle the problem asks you to find. Then, choose a formula that uses the given angles and side lengths and the ones you're looking for.
For example, if you have SAS (side, angle, side) and want the third side, or if you have SSS (side, side, side) and need an angle, use the Law of Cosines. If you have two sides and one angle, but none of the sides are opposite the given angle, you must use the Law of Cosines. If you have two angles and one side, you can use basic addition and subtraction to find the third angle because a triangle's interior angles always add up to 180 degrees, and then you can use the Law of Sines to find the remaining sides.
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When you want to find the measure of angles of non-right triangles
The Law of Sines and the Law of Cosines are used to determine the length of sides or measure of angles of triangles that are not right triangles. The Law of Sines is based on proportions and can be used to find the unknown angle or side of a triangle. This law can be used if certain combinations of measurements of a triangle are given.
There are two criteria that will provide a unique solution:
- ASA Criteria: Given two angles and the included side, to find the unknown side.
- AAS Criteria: Given two angles and a non-included side, to find the unknown side.
The Law of Sines can be used to solve oblique triangles, which are non-right triangles. According to the Law of Sines, the ratio of the measurement of one of the angles to the length of its opposite side equals the other two ratios of angle measure to opposite side.
The Law of Cosines can be used when you have three sides and any one angle, or when you have SAS (side-angle-side) and want the third side, or if you have SSS (side-side-side) and need an angle. Each of the three laws of cosines begins with the square of an unknown side opposite a known angle. To solve for a missing side measurement, the corresponding opposite angle measure is needed.
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Frequently asked questions
Use the Law of Sines when you have a side and an opposite angle and another side. You can then use sin inverse to find the angle.
No, in this case, you have to use the Law of Cosines.
Yes, the Law of Sines works for any triangle.
Use the Law of Cosines when you have the values of two sides and one non-included angle (SAS) or the measures of all three sides (SSS) of a triangle.
No, the Law of Cosines is for oblique triangles, which means triangles that do not have a 90-degree angle.
