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Law of Sines. Law of Cosines.

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Law of Sines

The Law of Sines, also known as the Sine Rule, is a mathematical rule used to solve triangles that are not right-angled. It relates the lengths of the sides of a triangle to the sine of their opposite angles. Consider a triangle with sides of lengths \(a\), \(b\), and \(c\), and angles opposite those sides of measures \(A\), \(B\), and \(C\), respectively.
The Law of Sines states that: \( \frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C} \)

In other words, the ratio of the length of each side to the sine of its opposite angle is the same for all sides. This ratio is often denoted by the symbol \(R\) and is called the circumradius of the triangle. Therefore, the Law of Sines can also be expressed as:
\( \frac{a}{R} = \frac{b}{R} = \frac{c}{R} \)

The Law of Sines can be used to find the length of a side or the measure of an angle of a triangle, given the lengths of two sides and the measure of the angle opposite one of them, or given the lengths of two sides and the measures of the angles opposite them.
To find the length of a side, say side \(a\), we can use the formula:
\( a = R \cdot sin A \)

To find the measure of an angle, say angle \(A\), we can use the formula:
\( sin A = \frac{a}{R} \)

It is important to note that the Law of Sines can only be used in non-right triangles, and it may not always provide a unique solution. In particular, if two sides and their included angle are known, there may be two possible triangles that can be formed. In such cases, the Law of Cosines can be used to determine which of the two solutions is correct.

Here are some additional points to keep in mind while working with the Law of Sines:

Ambiguity: As mentioned earlier, there can be an ambiguity in using the Law of Sines, which arises when two triangles can be formed with the given information. This is known as the ambiguous case or the case of two triangles. In this case, we need to use additional information, such as the lengths of another side or the measures of another angle, or we need to use the Law of Cosines to determine which of the two possible solutions is correct.

Range of values: The sine function has a range between -1 and 1, so the ratio of a side length to its opposite angle's sine value must be a positive number. This limits the range of possible values for the angles and sides of the triangle. Specifically, the largest angle of a triangle must be less than or equal to 90 degrees, and the ratio of the length of any side to the sine of its opposite angle must be greater than or equal to 2 (otherwise, the angle would be too large for the given side lengths).

Units: The units of measurement used for the sides and angles must be consistent. For example, if the side lengths are measured in centimeters, then the angles must be measured in radians or degrees. It is important to convert between different units of measurement as needed to ensure consistency.

Applications: The Law of Sines is used in a variety of applications, such as navigation, surveying, and engineering. It can be used to determine the height of an object, the distance between two points, or the angle of inclination of a slope.

Overall, the Law of Sines is a powerful tool for solving non-right triangles and has important applications in various fields. However, it is important to be aware of its limitations and potential ambiguities when working with it.

Law of Cosines

The Law of Cosines is a fundamental theorem in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is especially useful for solving triangles when the angle between two known sides is given, or when all three sides are known and you want to find the angles. The Law of Cosines is a generalization of the Pythagorean theorem, which is a special case for right-angled triangles.

For any triangle with sides of length \(a\), \(b\), and \(c\), and angles opposite to these sides denoted by \(A\), \(B\), and \(C\), respectively, the Law of Cosines states:
\(c^2 =a^2 + b^2 - 2ab \cdot cosC \)

You can also express the Law of Cosines for the other two sides:
\( a^2 = b^2 + c^2 - 2bc \cdot cosA \)
\( b^2 = a^2 + c^2 - 2ac \cdot cosB \)

Here's a brief outline of the steps to solve a triangle using the Law of Cosines:

Identify the given information: You should have at least three pieces of information about the triangle, including at least one side length.

Determine the appropriate form of the Law of Cosines: Based on the given information, choose the version of the Law of Cosines that will help you find the missing side length or angle.

Substitute the known values: Plug the given values into the chosen formula.

Solve for the unknown: Use algebraic techniques and/or a calculator to find the missing side length or angle.

Repeat steps 2-4 as necessary: Continue using the Law of Cosines or other trigonometric relationships (such as the Law of Sines) to find the remaining unknown side lengths and angles.

The Law of Cosines is an essential tool for solving oblique triangles (triangles without a right angle), as it allows you to find unknown side lengths and angles given sufficient information. It also provides a way to verify whether a triangle is acute, right, or obtuse.
If \(c^2 < a^2 + b^2 \), the triangle is acute.
If \(c^2=a^2 + b^2 \), the triangle is right-angled.
If \( c^2> a^2 + b^2 \), the triangle is obtuse.