Cecily Zamora
The Law of Cosines is a trigonometric rule that can be used to solve oblique triangles when two sides and the included angle are known. In some cases, the Law of Sines may be easier to apply, but it can lead to an ambiguous case where there are two possible angles, and both must be checked to see if they produce a solution. The Law of Cosines, on the other hand, involves solving a quadratic equation, and each positive solution of the equation corresponds to a valid triangle. This means that while the Law of Cosines may not have an ambiguous case in the same way as the Law of Sines, it can still result in multiple valid triangles or no triangle at all, depending on the values of the discriminant in the quadratic equation.
| Characteristics | Values |
|---|---|
| Law of Sines vs. Law of Cosines | The Law of Sines requires fewer calculations than the Law of Cosines, but the Law of Cosines uses only the original values, instead of the results of previous calculations and approximations. |
| When to Use Each | If we are solving a right triangle, we don't need the Laws of Sines and Cosines; all we need are the definitions of trigonometric ratios. For oblique triangles, the Law of Sines is easier to apply in the ambiguous case, but there will be two possible angles, and we must check each angle to see if it produces a solution. |
| Using the Law of Cosines in the Ambiguous Case | The Law of Cosines involves solving a quadratic equation, and each positive solution of the equation yields a solution of the triangle. The quadratic formula will tell us how many triangles have the given properties: one positive solution means one triangle, two positive solutions mean two triangles, and no positive solutions mean there is no triangle with the given properties. |
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What You'll Learn
The Law of Sines is easier to apply in the ambiguous case
The Law of Cosines can also be used to solve the ambiguous case, but it involves solving a quadratic equation. The quadratic formula will tell us how many triangles have the given properties. If the quadratic equation has one positive solution, there is one triangle. If the equation has two positive solutions, there are two triangles. If there are no positive solutions, there is no triangle with the given properties.
The Law of Sines requires fewer calculations than the Law of Cosines, as the latter uses only the original values, while the former involves results from previous calculations and approximations. Each additional calculation introduces inaccuracies, so using given values is preferable. Therefore, for the sake of accuracy and ease of calculation, the Law of Sines is the preferred method in the ambiguous case.
It is worth noting that the Law of Sines is not always the best choice. For example, when we know two sides of a triangle and the included angle, the Law of Cosines is more suitable. Additionally, if we have two angles and one side, basic addition and subtraction can be used to find the third angle, and the Law of Sines can be applied subsequently.
In summary, while the Law of Sines is generally easier to apply in the ambiguous case, the Law of Cosines may be more appropriate in certain situations, depending on the given information and the specific problem at hand.
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The Law of Cosines involves solving a quadratic equation
The Law of Cosines is a generalization of the Pythagorean theorem that applies to all triangles, regardless of the size of angle C. It is particularly useful when we know two sides of a triangle and the included angle, or when we know all three sides of a triangle.
For example, let's consider a triangle with sides a = 8, b = 3, and angle B = 14.4°. Using the Law of Cosines, we can find the third side of the triangle by solving the quadratic equation:
$$
\begin{aligned}
B^2 &= a^2 + c^2 - 2ac \cos B \\
3^2 &= 8^2 + c^2 - 2(8)c\cos 14.4^{\circ} \\
9 &= 64 + c^2 - 16c(0.9686) \\
0 &= c^2 - 15.497c + 55 \\
C &= \frac{15.497 \pm \sqrt{(-15.497)^2 - 4(1)(55)}}{2(1)} \\
C &= \frac{15.497 \pm 4.490}{2} = 5.503 \text{ or } 9.994
\end{aligned}
$$
Since there are two positive solutions for side c, there are two triangles with the given properties. We can then apply the Law of Cosines again to find angle C in each triangle.
The Law of Cosines is a powerful tool for solving triangles, especially in ambiguous cases, but it involves solving quadratic equations, which can be more complex than the basic trigonometric ratios used in right triangles.
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The ambiguous case can be avoided on the SSA triangle
When solving for a right triangle, the Laws of Sines and Cosines are not required; instead, trigonometric ratios are used. However, for oblique triangles, we can identify the ambiguous case, also known as the SSA case, where we know two sides and the angle opposite one of them. In this case, the Law of Sines is easier to apply, but it results in two possible angles, and we must check each angle to determine if it produces a solution. This is because there are always two angles with a given sine value. To avoid this ambiguity, the Law of Cosines can be used, which involves solving a quadratic equation. Each positive solution of the quadratic equation corresponds to a solution for the triangle.
For example, let's consider a triangle with angle \(B = 14.4^\circ\) and sides \(a = 8\) and \(b = 3\). We can use the Law of Sines to find angle \(B\) as follows:
$$
\begin{align*}
\frac{\sin(B)}{6} &= \frac{\sin(40^\circ)}{10} \\
\sin(B) &= 0.3857
\end{align*}
$$
This is the ambiguous case, as there are two angles between \(0^\circ\) and \(180^\circ\) with a sine value of \(0.3857\). To find the correct angle, we need to check each possibility.
By using the Law of Cosines, we can avoid this ambiguity. The Law of Cosines states that for a triangle with sides \(a\), \(b\), and \(c\) and angles \(A\), \(B\), and \(C\), the following equations hold true:
$$
\begin{align*}
A^2 &= b^2 + c^2 - 2bc \cos(A) \\
B^2 &= a^2 + c^2 - 2ac \cos(B) \\
C^2 &= a^2 + b^2 - 2ab \cos(C)
\end{align*}
$$
In our example, we can use the first equation to find the third side of the triangle:
$$
\begin{align*}
A^2 &= b^2 + c^2 - 2bc \cos(A) \\
8^2 &= 3^2 + c^2 - 2 \cdot 3c \cdot \cos(14.4^\circ) \\
64 &= 9 + c^2 - 5.76c \\
55 &= c^2 \\
\sqrt{55} &= c
\end{align*}
$$
So, the third side of the triangle is approximately \(c \approx 7.42\).
By using the Law of Cosines, we avoided the ambiguous case and directly solved for the unknown side of the triangle. This approach ensures that we find the correct solution without having to check multiple possibilities, making it a more efficient method for solving oblique triangles in the SSA case.
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The Law of Cosines uses only the original values
The Law of Cosines is a trigonometric rule that relates the length of a triangle to the cosines of one of its angles. It is used to find the length of the third side of a triangle when the length of the other two sides and the angle between them are known. The formula for this is:
A^2 = b^2 + c^2 - 2bc cos(A)
Where a, b, and c are the sides of the triangle, and A is the angle between sides b and c.
For example, if we know two sides of a triangle, a and b, and the acute angle α opposite one of them, there may be one solution, two solutions, or no solution, depending on the size of a in relation to b and α. If we use the Law of Sines for the ambiguous case, we must check whether both possible angles result in a solution.
The Law of Cosines is also useful when we know all three sides of a triangle and want to find an angle. In this case, we can use the formula:
Cos α = (b^2 + c^2 - a^2) / (2bc)
The Law of Cosines is a generalization of the Pythagorean theorem, which only holds for right triangles. It can be used to solve oblique triangles, where the Law of Sines is not helpful.
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The quadratic formula will tell us how many triangles have the given properties
When using the Law of Cosines to find a side in the ambiguous case, the quadratic formula will tell us how many triangles have the given properties. The quadratic formula can be used to find the solutions, or roots, of quadratic equations.
A quadratic equation can have one solution, two solutions, or no solution, depending on the value of the discriminant, $b^2 - 4ac$. Each positive solution of the equation yields a solution of the triangle. If the quadratic equation has one positive solution, there is one triangle. If the quadratic equation has two positive solutions, there are two triangles. If the quadratic equation has no positive solutions, there is no triangle with the given properties.
For example, let's consider a triangle with angle $B = 14.4^\circ$ and sides $a = 8$ and $b = 3. Using the Law of Cosines, we can find the third side of the triangle. The Law of Cosines states that if the angles of a triangle are $A, B,$ and $C$, and the opposite sides are respectively $a, b,$ and $c$, then:
$$
\begin{align*}
A^2 &= b^2 + c^2 - 2bc \cos(A) \\
B^2 &= a^2 + c^2 - 2ac \cos(B) \\
C^2 &= a^2 + b^2 - 2ab \cos(C)
\end{align*}
$$
By substituting the given values and solving the quadratic equation, we can find the value of $c$ and determine the number of triangles with the given properties.
It is worth noting that the Law of Sines is easier to apply in the ambiguous case, but it may result in two possible angles, and we must check each angle to see if it produces a solution. On the other hand, the Law of Cosines uses only the original values, which can lead to more accurate calculations.
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Frequently asked questions
The Law of Cosines is a generalization of the Pythagorean theorem that applies to all triangles, regardless of the size of angle C. It states that if the angles of a triangle are A, B, and C, and the opposite sides are a, b, and c, then:
a^2 = b^2 + c^2 - 2bc * cos(A)
b^2 = a^2 + c^2 - 2ac * cos(B)
c^2 = a^2 + b^2 - 2ab * cos(C)
The Law of Cosines is used when we know two sides of a triangle and the included angle. It is also used to find an angle when we know all three sides of a triangle.
The Law of Cosines itself does not have an ambiguous case. However, it can be used to resolve an ambiguous case that arises when using the Law of Sines. An ambiguous case occurs when there are two possible angles that satisfy the given conditions, and we must check each angle to see if it produces a solution.
To resolve an ambiguous case, we can use the Law of Cosines to find the third side of the triangle. This involves solving a quadratic equation, and each positive solution of the equation yields a solution for the triangle. If there is one positive solution, there is one triangle. If there are two positive solutions, there are two triangles. If there are no positive solutions, there is no triangle with the given properties.
