Gavin Howe
The cosine rule is a trigonometric rule used to determine angle measures or side lengths within non-right triangles. When using the cosine rule, there seems to be no ambiguous case. This is because the cosine rule requires all sides to be given or for the SAS angles to be given, both of which are unambiguous cases. On the other hand, the sine rule can result in an ambiguous case, where there are two possible triangles or no triangle at all. This occurs when there are two valid solutions for a situation, or when the given measurements result in no triangle.
| Characteristics | Values |
|---|---|
| Ambiguous case | When the relationship between sides and an angle does not produce a triangle or results in two possible triangles |
| Cosine rule | Used to determine angle measure or side lengths within non-right triangles |
| Sine rule | Used to determine angle measure or side lengths within right triangles |
| SSA triangle | When the given angle is opposite to the smaller of the two given sides, there will be an ambiguous case |
| Sine law | Can result in two different values for angle measures |
| Cosine law | Does not result in two different values for angle measures |
| SAS | Angles given |
| SSS | All sides given |
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What You'll Learn
The Cosine Rule and its applications
The law of cosines, also known as the cosine formula or cosine rule, is a fundamental concept in trigonometry. It relates the lengths of the sides of a triangle to the cosine of one of its angles. The rule is particularly useful for solving triangles when all three sides or two sides and the included angle are known.
The cosine rule can be expressed by the formula:
C^2 = a^2 + b^2 - 2ab cos(C)
In this formula, a, b, and c represent the lengths of the sides of a triangle, and C is the angle between sides a and b. By rearranging the formula, it can also be written in terms of a^2 or b^2. This rule is a generalisation of the Pythagorean theorem, which only applies to right-angled triangles.
The cosine rule has been utilised by mathematicians for centuries, dating back to ancient times. For instance, an equivalent to the spherical law of cosines was used by 9th-century mathematicians al-Khwārizmī and al-Battānī. In the 11th century, al-Bīrūnī applied the law of cosines to solve astronomical problems, and it later appeared in Europe in the 15th century in Regiomontanus's "De triangulis omnimodis."
The cosine rule is particularly useful in solving triangles, especially when dealing with non-right triangles. It can be used to find the third side of a triangle when two sides and the angle between them are known, or when two sides and an angle opposite to one of them are given. However, it's important to note that there can be ambiguous cases when applying the cosine rule. For example, with SSA triangles, if the given angle is opposite to the smaller of the two given sides, there may be an ambiguous case.
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The Law of Sines vs. the Law of Cosines
The Law of Sines and the Law of Cosines are both used to determine angle measures or side lengths within non-right triangles. However, the application of each law depends on the given information.
The Law of Sines is used when two sides and one non-included angle are known. In this case, the formula needs to be rearranged, and the last step is always taking the inverse ratio to get the measure of the angle. It is important to avoid using sines of obtuse angles where possible, as the function is cyclic.
On the other hand, the Law of Cosines is used when the lengths of two sides and the angle between them are known. By using the Pythagorean Theorem, the Law of Cosines can be expressed in expanded form.
It is worth noting that both laws can be used together to solve certain problems. For example, when solving for an angle using the Law of Sines, the Law of Cosines can be used to determine the corresponding side length.
In terms of ambiguous cases, the Law of Sines can yield ambiguous results, leading to two possible triangles. This occurs when the relationship between the sides and an angle does not produce a unique triangle. On the other hand, the Law of Cosines does not seem to have an ambiguous case when used with SSS (all sides given) or SAS (side-angle-side given) triangles. However, with SSA triangles (two sides and the included angle given), both the Law of Sines and the Law of Cosines can yield two possible triangles, resulting in an ambiguous case.
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The ambiguous case of the Sine Law
For example, if we know two sides of a triangle and the angle opposite one of them, there may be one solution, two solutions, or no solution, depending on the size of the sides and the angle. If the angle is obtuse, there is one solution if the side opposite the angle is longer than the other side, and no solution if the side opposite the angle is shorter or equal in length to the other side.
The Sine Law is not helpful when we know two sides of a triangle and the included angle, as it can result in an ambiguous case. In this case, we need to use the Cosine Law, which does not have an ambiguous case. The Cosine Law requires all the information of SAS (two sides and the included angle) or SSS (all three sides), which fully specifies the triangle.
The absence of an ambiguous case in the Cosine Law can be attributed to the fact that it uses only the original values, whereas the Sine Law uses the results of previous calculations and approximations, introducing inaccuracies that grow with each additional calculation.
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Using the Cosine Rule to solve for angles
The Law of Cosines helps us solve some triangles. When solving for an angle, the formula needs to be rearranged, and the last step is always taking the inverse ratio to get the measure of the angle.
The formula for the Law of Cosines is:
C^2 = a^2 + b^2 - 2ab cos(C)
Where 'c' is the side opposite angle C, and 'a' and 'b' are the other two sides of the triangle.
For example, let's say we know that angle C = 37 degrees, side 'a' = 8, and side 'b' = 11. We can plug these values into the formula:
C^2 = 8^2 + 11^2 - 2 * 8 * 11 * cos(37)
C^2 = 64 + 121 - 176 * 0.798
C^2 = 44.44
To find the value of 'c', we take the square root of 44.44, which gives us c = 6.67 to 2 decimal places.
It's worth noting that the cosine of an obtuse angle is always negative.
Now, let's discuss ambiguous cases. In some cases, the relationship between the sides and an angle may not produce a triangle, or they may result in two possible triangles. This is called an ambiguous case. With the Cosine Rule, there doesn't seem to be an ambiguous case. This may be due to the fact that the Cosine Rule requires you to have all sides given or the SAS (side-angle-side) angles given, which are unambiguous cases.
However, one user points out that when you end up with a quadratic equation, you do have an ambiguous case because quadratic equations generally have two solutions. These two solutions will correspond to the two possible triangles with the given information.
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The Cosine Rule for fully-specified triangles
The Cosine Rule, also known as the Law of Cosines, is a formula used to solve triangles. It can be used in any triangle where you are trying to relate all three sides to one angle. The formula is:
C^2 = a^2 + b^2 - 2ab cos(C)
Where 'a' and 'b' are the lengths of two sides of the triangle, 'c' is the length of the third side, and 'C' is the angle between sides 'a' and 'b'.
The Cosine Rule is particularly useful when we are given all three sides of a triangle and no angles, and we want to find an angle. It can also be used when we are given two sides and a contained angle and want to find the side length that corresponds to the given angle.
It's important to note that the Cosine Rule does not have an ambiguous case when used with fully-specified triangles. This means that if you have all three sides of a triangle (SSS) or two sides and the included angle (SAS), rearranging the sides will always yield the same triangle.
However, it's worth mentioning that the Cosine Rule can still provide two solutions when used with SSA triangles (two sides and the non-included angle). In such cases, the two solutions correspond to the two possible triangles that can be formed with the given information.
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Frequently asked questions
The Cosine Rule can only be applied when the conditions are exactly those where the ambiguous case no longer applies.
When solving for an angle, the formula needs to be rearranged and the last step is always taking the inverse ratio to get the measure of the angle.
We can use the Law of Cosines to solve the ambiguous case.
We use the Cosine Law when we need to determine angle measure or side lengths within non-right triangles.
