Cecily Zamora
The law of sines is a trigonometric rule used to determine the relationship between the sides and angles of a triangle. It is used to find the remaining sides of a triangle when given two angles and a side or two sides and a non-enclosed angle. However, the law of sines cannot be used as the first step when solving an SAS (Side-Angle-Side) triangle, where the lengths of two sides and the measure of the angle between them are known. This is because the law of sines requires an angle-side opposite pair, which is not available in an SAS triangle. Instead, the law of cosines is used to find the third side in an SAS triangle, and the law of sines can then be applied to determine the remaining angles.
| Characteristics | Values |
|---|---|
| SAS triangle | Side-Angle-Side triangle where the lengths of two sides and the measure of the angle between them are known |
| Law of Sines | Used to compute the remaining sides of a triangle, given two angles and a side or two sides and one of the non-enclosed angles |
| Applicability to SAS triangle | Cannot be used as the first step to solve an SAS triangle due to the lack of an angle-side opposite pair; Law of Cosines is used instead |
| Angle-side opposite pair | The length of one side and the measure of the angle across from it, which is necessary for the application of the Law of Sines |
| Ambiguous case | Occurs when there are two possible solutions to a triangle due to the possibility of creating two different triangles with the given information |
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What You'll Learn
The Law of Sines formula
The Law of Sines, also known as the Sine Rule, is a trigonometric formula used to find unknown sides or angles in a triangle. It is particularly useful when dealing with non-right-angled triangles. 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, regardless of which angle-side pair is chosen. This principle can be expressed by the equation:
${\displaystyle {\frac {a}{\sin {\alpha }}}\,=\,{\frac {b}{\sin {\beta }}}=\,{\frac {c}{\sin {\gamma }}}}$
Where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the corresponding opposite angles.
The Law of Sines can be used to solve for unknown sides or angles in a triangle when certain information is known. For example, if two angles and a side are known, the law can be applied to find the remaining sides. This technique is known as triangulation.
When solving an SAS triangle (a triangle in which two sides and the included angle are known), the Law of Sines is used to find the angle opposite the shortest of the two given sides. This ensures that the angle found is acute, preventing ambiguity in the solution.
It's worth noting that the Law of Sines has different forms depending on the geometric context. For example, in spherical geometry, the Law of Sines deals with triangles on a sphere, where the sides are arcs of great circles, and the formulas involve additional considerations. Similarly, in hyperbolic geometry, the Law of Sines takes on a modified form to account for the curvature of -1.
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The Law of Cosines formula
The Law of Cosines, also known as the Cosine Formula or Cosine Rule, is a formula in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides a, b, and c, and angles A, B, and C, the Law of Cosines can be used to find the unknown side of a triangle when two sides and their included angle are given (i.e., for SAS or SSA triangles).
The formula is as follows:
C^2 = a^2 + b^2 - 2ab cos(C)
A^2 = b^2 + c^2 - 2bc cos(A)
B^2 = a^2 + c^2 - 2ac cos(B)
Where c is the unknown side, and a, b, and C are known. This formula can be derived from the Pythagorean theorem and can be used to solve triangles that are not right triangles, making it a generalization of the Pythagorean theorem.
When solving an SAS triangle, the Law of Cosines can be used to find the side opposite the given angle. However, the Law of Sines is typically preferred in such cases because it is easier to apply and does not suffer from the same round-off errors as the Law of Cosines when the triangle is very acute. By using the Law of Sines, we can guarantee that the angle we find will be acute, avoiding any ambiguity that might arise from obtaining multiple solutions.
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Using the Law of Sines to find the angle opposite the shortest remaining side
The law of sines, also known as the sine rule, is a mathematical principle that is used to find the unknown side or angle in a triangle. This law is applied when certain combinations of measurements of a triangle are provided, such as two angles and the included side, or two angles and a non-included side. The sine rule can be expressed as:
$$\co: 11>\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
Where $a, b$, and $c$ are the sides of a triangle, and $A, B$, and $C$ are the angles.
When solving an SAS (side-angle-side) triangle, the law of sines is used to find the angle opposite the shortest remaining side. This is done to ensure that the angle found is either the smallest or the second smallest angle in the triangle, avoiding ambiguity. By choosing the angle opposite the shorter side, we guarantee that this angle is not the largest in the triangle and, therefore, must be acute. This means that the angle will always be less than $90^\circ$.
The law of cosines can also be used to find the angle in an SAS triangle. However, the law of sines is often preferred due to its relative ease of use and the fact that it provides a single, unambiguous answer.
- Use the law of cosines to find the side opposite the given angle.
- Apply the law of sines to determine the angle opposite the shorter of the two given sides. This angle will always be acute.
- Calculate the third angle by subtracting the measure of the given angle and the angle found in step 2 from $180^\circ$.
It is important to note that the sine function is positive in quadrants one and two, which can lead to two possibilities for the angle. However, by choosing the angle opposite the shorter side, we can ensure that we obtain the correct answer.
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Using the Law of Cosines to find the third side
The Law of Cosines is a useful tool for solving triangles when all three sides or two sides and their included angle are given. It can be used to find the third side of a triangle when two sides and the angle between them are known.
The formula for the Law of Cosines is:
C^2 = a^2 + b^2 − 2ab x cos(C)
Here, 'a' and 'b' are the two known sides, and 'C' is the angle between them. By inputting the values of the known sides and angle into this formula, we can find the length of the third side, denoted as 'c'.
For example, let's say we have a triangle with side 'a' measuring 8 units, side 'b' measuring 11 units, and the angle between them, angle 'C', measuring 37 degrees. We can use the Law of Cosines to find the length of the third side:
C^2 = 8^2 + 11^2 − 2 x 8 x 11 x cos(37°)
C^2 = 64 + 121 − 176 x 0.798...
C^2 = 44.44...
C = √44.44 = 6.67 to 2 decimal places
So, the length of the third side is approximately 6.67 units.
It's worth noting that the Law of Sines can also be used to find the third side of a triangle in certain cases, specifically when two sides and the angle opposite one of them are known. However, the Law of Cosines provides a direct formula that is generally easier to use and avoids ambiguity in solutions.
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The ambiguous case
To determine if there is a second valid angle, follow these steps:
- Find the value of the unknown angle using the Law of Sines.
- Once you find the value of the angle, subtract it from 180° to find the possible second angle.
- To check if the second angle is valid, add the two angles found in steps 1 and 2. If their sum is less than 180°, a triangle can exist, and the second angle is valid. If the sum is greater than 180°, the second angle is not valid, as the three angles of a triangle must add up to 180°.
For example, let's consider a triangle with b = 10 inches and c = 6 inches. This triangle is a candidate for the ambiguous case, as we are given two sides and an angle not between them. Using the Law of Sines, we can find one value for angle B. To check if there is another possible value for angle B, we subtract this value from 180°. If the sum of the two possible values for angle B is less than 180°, both angles are valid, and two triangles can be formed.
It is important to note that the Law of Sines is used to find the angle opposite the shorter of the two given sides in an SAS triangle. This ensures that the angle found is acute and avoids ambiguity. By choosing the angle opposite the shorter side, there must be at least one angle greater than the angle being found, guaranteeing that it cannot be obtuse.
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Frequently asked questions
No, the law of sines cannot be used as the first step to solve an SAS triangle.
The law of sines requires an angle-side opposite pair, which is not provided in an SAS triangle.
An SAS triangle is a triangle where the lengths of two sides and the measure of the angle between them are known.
The law of cosines can be used to solve an SAS triangle.
The law of sines can be used when two angles and a side are given, or when two sides and a non-included angle are given.
