Anaya Nolan
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. On the other hand, the Law of Cosines is used when you know three sides and one angle, or two sides and one angle (SAS, SSS, or ASA). However, the Law of Cosines is generally preferred as it does not have an ambiguous case, unlike the Law of Sines, which can introduce extraneous solutions. In some cases, using the Law of Sines may require testing your results by substituting all sides and angles to ensure equivalent results.
| Characteristics | Values |
|---|---|
| When to use the Sine Law | When you have a side and an opposite angle and another side |
| When to use the Cosine Law | When you have two sides and one angle, but none of the sides are opposite to the given angle |
| When not to use the Sine Law | When you have two angles and one side, or when dealing with obtuse angles |
| When not to use the Cosine Law | When dealing with equilateral triangles |
| When to use both | When you know the lengths of sides a and b, and the measurement of the angle between them |
| Disadvantages of using the Sine Law | May introduce an ambiguous case and create extraneous solutions |
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What You'll Learn
When you have two sides and one non-adjacent angle
The law of sines and the law of cosines are used to solve for unknown sides or angles in a triangle. When you have two sides and one non-adjacent angle of a triangle, you can use the law of sines to solve for the unknown angle.
The law of sines states that the ratio of the length of one side of a triangle to the sine of its opposite angle is constant. In other words, when you divide side a by the sine of angle A, it is equal to side b divided by the sine of angle B, and also equal to side c divided by the sine of angle C.
To use the law of sines to solve for the unknown angle when you have two sides and one non-adjacent angle, you need to set up and solve for the equation using the given information. For example, if you know the lengths of sides a and b, and the measure of angle C, you can use the law of sines to find the measure of angle A.
It's important to note that the law of sines assumes that you are working with angles in degrees, not radians. Additionally, the law of sines may yield two possible solutions for the unknown angle, so it's important to check that the solution makes sense in the context of the problem.
In contrast, the law of cosines is used when you have two sides and one adjacent angle of a triangle. The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It can be used to find the length of a side or the measure of an angle, depending on the given information.
In summary, when you have two sides and one non-adjacent angle of a triangle, you can use the law of sines to solve for the unknown angle. This involves setting up and solving for the equation using the given side lengths and angle measures. The law of sines can provide valuable insights into the relationships between the sides and angles of a triangle.
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When you have three sides and want to find an angle
The Law of Cosines is used when you know the lengths of sides a and b, and you also know the measurement of the angle between these sides, which is angle C. In other words, you know three sides and want to find an angle.
For example, let's say you have a triangle with sides a = 17, b = 6, and c = 15. To find angle C, you can use the Law of Cosines formula:
$$C = \arccos\left( \frac{6^2 + 17^2 - 15^2}{2(6)(17)} \right)$$
Now, let's calculate the value of angle C:
$$C = \arccos\left( \frac{36 + 289 - 225}{102} \right) = \arccos\left( \frac{6}{102} \right) = \arccos(0.0588)$$
Using a calculator, the value of the above expression works out to be:
$$C \approx 60.65^\circ$$
So, angle C is approximately $60.65^\circ$.
It's important to note that the Law of Cosines is generally preferred over the Law of Sines when finding angles because the Law of Sines can introduce ambiguous cases and extraneous solutions. The Law of Cosines does not have this issue, so it is recommended to use it whenever possible.
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When you have two angles and the side between them
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.
It is important to note that the Law of Sines can be ambiguous and lead to extraneous solutions. Therefore, it is advisable to use the Law of Cosines whenever possible. The Law of Cosines is used when you know the lengths of two sides and the angle between them, and you want to find the third side.
In some cases, you may need to use both the Law of Sines and the Law of Cosines to solve a problem. For example, if you have a triangle with sides a = 17, b = 6, and c = 15, you can use the Law of Cosines to find the first angle and then the Law of Sines to find the other two angles.
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When you have SSS and need to find a side
The sine law lets you relate two sides and their two opposite angles, whereas the cosine law is used for three sides and any one angle. When you have SSS and need to find a side, you can use the law of cosines to find the angle between two sides.
For example, if you have sides a, b, and c, you can use the law of cosines to find the angle between sides a and b:
> γ = acos((a² + b² − c²)/(2ab))
You can then apply the law of cosines again to find the angle between sides a and c. Finally, you can compute the remaining angle using the fact that the angles of a triangle add up to 180 degrees:
> α = 180° - β - γ
It is important to note that the law of sines is difficult to use with angles above 90 degrees. Therefore, when using SSS to find a side, it is recommended to first find the largest angle using the law of cosines. This ensures that the other angles are acute (less than 90 degrees), allowing the law of sines to be used without difficulty.
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When you have SAS and want to find an angle
When solving a triangle with two sides and one angle (SAS), the law of sines is used to find the angle opposite the shorter of the two given sides. This is because the angle opposite the shorter side will always be acute, meaning it cannot be obtuse. This guarantees that there will be no ambiguity when using inverse sine to find the angle.
The law of cosines can also be used to find the angle in an SAS triangle. However, the law of sines is generally preferred in this case because it is perceived to be easier, with fewer operations to perform.
To use the law of sines to find the angle in an SAS triangle, follow these steps:
- Identify the shorter of the two given sides.
- Use the law of cosines to find the side opposite the given angle.
- Apply the law of sines to find the angle opposite the shorter side identified in step 1.
- Determine the third angle by subtracting the measure of the given angle and the angle found in step 3 from 180 degrees.
It is worth noting that the law of tangents (or a variant) can also be used to solve for the smaller angle directly, without first calculating the other side.
In summary, when you have SAS and want to find an angle, the law of sines is typically the preferred choice due to its ease of use and the absence of ambiguity in the solution.
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Frequently asked questions
When there are no known angles or sides. Ideally, the formula you choose should only have one unknown quantity.
Yes, but you have to be careful because sin(x) = sin(180-x), so your answer will always be an acute angle. If you know the angle is obtuse, use 180-x.
The law of sines is a mathematical equation relating the lengths of the sides of any triangle to the sines of its angles.
The spherical law of sines is credited to 10th-century scholars Abu-Mahmud Khujandi and Abū al-Wafā. It was given prominence by Abū Naṣr Manṣūr and proved by 13th-century mathematician Naṣīr al-Dīn al-Ṭūsī.
If your triangle is a right triangle, use the law of cosines. Otherwise, use the law of sines.
