Gavin Howe
The law of cosines helps us solve some triangles. The formula for the law of cosines is c^2 = a^2 + b^2 - 2ab cos(C). The cases of obtuse triangles and acute triangles are treated separately, with the cosine of an obtuse angle always being negative. Using algebraic measures, the two cases can be treated simultaneously. The law of cosines can be proven algebraically from the law of sines and trigonometric identities.
| Characteristics | Values |
|---|---|
| Formula | a2 + b2 - 2ab cos(C) = c^2 |
| Acute angle case | CK < 0 |
| Obtuse angle case | CK > 0 |
| Acute triangles | Correspond to the case of positive cosine |
| Obtuse triangles | Correspond to the case of negative cosine |
| Round-off errors | Produced in floating-point calculations for acute triangles |
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What You'll Learn
Acute triangles
The law of cosines, also known as the cosine formula or cosine rule, is used to solve triangles. It is useful when all three sides are given or when two sides and their included angle are given. The law of cosines can be used to find the third side of a triangle when the lengths of the other two sides and the angle between them are known.
The law of cosines can be used to solve acute triangles. In the case of acute triangles, the law of cosines is used to find the unknown side of a triangle when the length of the other two sides is given and the angle between them is known. The formula for this is:
A^2 = b^2 + c^2 - 2bc cos(α)
Where a is the unknown side, b and c are the known sides, and α is the angle between b and c.
For example, let's say we have a triangle ABC with sides a = 10cm, b = 7cm, and c = 5cm. We can use the law of cosines to find the measure of angle x. Using the formula:
A^2 = b^2 + c^2 - 2bc cos(α)
We can substitute the given values:
10^2 = 7^2 + 5^2 - 2(7)(5) cos(x)
Simplifying the equation gives:
100 = 49 + 25 - 70 cos(x)
Subtract 74 from both sides:
26 = -70 cos(x)
Divide by -70:
Cos(x) = -26/70
Using a calculator, we find that x is approximately equal to 140.6 degrees.
The law of cosines can also be used to find the angles of an acute triangle when all three sides are known. This is done using the formula:
C^2 = a^2 + b^2 - 2ab cos(C)
For example, let's say we have a triangle with sides a = 8, b = 11, and angle C = 37 degrees. We can use the law of cosines to find the length of the third side, c.
C^2 = 8^2 + 11^2 - 2(8)(11) cos(37)
C^2 = 64 + 121 - 176 cos(37)
C^2 = 185 - 140.72
C^2 = 44.28
Taking the square root:
C = ±6.66
Since side lengths cannot be negative, we take the positive square root, so c is approximately equal to 6.66.
In summary, the law of cosines can be used to solve acute triangles by finding the unknown side or angle when the other two sides and one angle, or all three sides, are given.
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Obtuse triangles
The law of cosines, also known as the cosine formula or cosine rule, relates the lengths of the sides of a triangle to the cosine of one of its angles. It is used to determine the unknown sides of a triangle when the length of the other two sides and the angle between them are known. The formula for the law of cosines is:
> a^2 = b^2 + c^2 - 2bc cos α
Where a, b, and c are the sides of the triangle, and α is the angle between sides b and c.
The law of cosines can be used to solve for obtuse triangles, where one of the angles is greater than 90 degrees. In this case, the cosine of the obtuse angle is negative. The formula for the law of cosines still applies, but the negative sign of the cosine value must be considered.
For example, let's consider an obtuse triangle ABC with sides a = 10 cm, b = 7 cm, and c = 5 cm. We can use the law of cosines to find the angle x between sides a and b:
> a^2 = b^2 + c^2 - 2bc cos α
> 10^2 = 7^2 + 5^2 - 2(7)(5) cos x
> 100 = 49 + 25 - 70 cos x
> 100 = 74 - 70 cos x
> 26 = 70 cos x
> cos x = 26/70
> cos x = 0.371
Using a calculator or the inverse cosine function, we can find that the angle x is approximately 71.2 degrees.
It's important to note that when using the law of cosines for obtuse triangles, the negative sign of the cosine value is crucial. This negative sign arises because the cosine of an obtuse angle is always negative. This distinguishes obtuse triangles from acute triangles, where the cosine of the acute angle is positive.
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Round-off errors
The law of cosines, also known as the cosine formula or cosine rule, relates the lengths of the sides of a triangle to the cosine of one of its angles. It is used to determine the unknown side of a triangle when the length of the other two sides is given, along with the angle between them.
The law of cosines is useful for solving triangles when all three sides or two sides and their included angle are given. The formulae, however, produce high round-off errors in floating-point calculations if the triangle is very acute. In other words, if the side opposite the angle is small relative to the other sides, or if the angle itself is small, the law of cosines will yield a round-off error.
For example, consider a triangle with sides a = 17, b = 6, and c = 15. Using the law of cosines to find the first angle, and then the law of sines to find the other two, yields the following results:
C = arccos((6^2 + 17^2 - 15^2) / 2(6)(17)) = 60.647 degrees
B = arcsin(6 sin C / 15) = 20.405 degrees
A = arcsin(17 sin B / 6) = 81.051 degrees
The sum of these angles should be 180 degrees, but the result is 162 degrees. This is a significant error that challenges the validity of the laws.
One way to reduce such errors is to convert the cosine function into 1-2sin^2(A/2) when A is small. This method works most of the time.
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Trigonometric identities
The law of cosines, also known as the cosine formula or cosine rule, is a formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It can be used to solve for the third side of a triangle when two sides and the angle between them or two sides and an angle opposite to them are known. The formula is represented as:
$$a^2 + b^2 - 2ab \cos(C) = c^2$$
The law of cosines can be proven algebraically from the law of sines and standard trigonometric identities. Trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables. They are useful for simplifying expressions involving trigonometric functions and integrating non-trigonometric functions.
The Pythagorean identity, for example, is a fundamental relationship between the sine and cosine functions:
$$x^2 + y^2 = 1$$
This identity can be used to express one trigonometric function in terms of another:
$$\sin \theta = \pm \sqrt{1 - \cos^2 \theta}$$
$$\cos \theta = \pm \sqrt{1 - \sin^2 \theta}$$
The law of cosines can also be proven geometrically by calculating areas. The change in sign as the angle γ becomes obtuse requires a distinction between the acute and obtuse cases. The acute case involves a heptagon that can be cut into smaller pieces in two different ways to yield a proof of the law of cosines. The obtuse case involves a similar approach, considering the areas of parallelograms formed by the sides of the triangle.
The law of cosines has applications in solving triangles, particularly when finding the unknown sides of a triangle given certain information about its angles and known sides. It is a generalization of the Pythagorean theorem, which only applies to right triangles.
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Triangle congruence
The law of cosines relates the lengths of a triangle's sides to the cosine of one of its angles. Specifically, it helps determine the third side of a triangle when the lengths of the other two sides and the angle between them are known. This is achieved through the following formula:
A^2 = b^2 + c^2 - 2bc x cos(alpha)
Where 'a' is the unknown side, 'b' and 'c' are the known sides, and 'alpha' is the angle between the known sides.
For example, let's consider a triangle with sides a = 10cm, b = 7cm, and c = 5cm. We can use the law of cosines to find the measure of angle 'x' by inputting the given values into the formula:
10^2 = 7^2 + 5^2 - 2 x 7 x 5 x cos(x)
100 = 49 + 25 - 70 x cos(x)
100 - 74 = -70 x cos(x)
26 = -70 x cos(x)
26/70 = cos(x)
X = arccos(-26/70)
Therefore, the measure of angle 'x' in this triangle is approximately 125.3 degrees.
The law of cosines can also be used to find the angles of a triangle when all three sides are known. This is achieved by rearranging the formula to isolate the angle:
Cos(alpha) = (b^2 + c^2 - a^2) / (2bc)
By inputting the values of the sides and calculating the cosine inverse, we can find the measure of the angle.
The law of cosines is applicable to both acute and obtuse triangles, with the cosine of an obtuse angle always being negative. This distinction is important when applying the law of cosines to solve for triangle sides or angles.
In summary, the law of cosines is a valuable tool for solving triangles and determining triangle congruence. It allows us to find unknown sides or angles of a triangle when given sufficient information. By using the appropriate formula and performing the necessary calculations, we can solve for the unknown values and establish triangle congruence if the corresponding sides and angles are equal.
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Frequently asked questions
The law of cosines solves for two cases: the case of obtuse triangles and the case of acute triangles.
The formula for the law of cosines is:
(for all triangles)a^2 + b^2 − 2ab cos(C) = c^2
The law of cosines can be used to solve for sides in a triangle when given the measurements of the angles and sides.
The cosine of an obtuse angle is always negative, whereas the cosine of an acute angle is positive.
