Law Of Cosines Correct Equation For Side Length S

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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's a powerful tool for solving triangles, especially when you don't have a right angle. This law is particularly useful when you know two sides and the included angle (SAS) or when you know all three sides (SSS) of a triangle. In these scenarios, the law of cosines allows you to calculate the remaining sides or angles. The law of cosines is a generalization of the Pythagorean theorem, which only applies to right triangles. In fact, when the angle in question is 90 degrees, the cosine term becomes zero, and the law of cosines simplifies to the Pythagorean theorem. This makes the law of cosines a versatile tool that can be used for any type of triangle. The standard formula for the law of cosines is expressed in three different forms, each focusing on a different side of the triangle. These forms are derived from the same basic principle but are rearranged to isolate the desired side length. Understanding these different forms allows you to choose the most efficient equation for your specific problem. For example, if you are solving for side 'a', you would use the form that isolates a². If you are solving for an angle, you would rearrange the formula to isolate the cosine of that angle. The law of cosines is widely used in various fields, including surveying, navigation, and engineering. Surveyors use it to calculate distances and angles in land measurement. Navigators use it to determine the position and course of ships and aircraft. Engineers use it in structural design and other applications where triangles and angles are involved. Mastering the law of cosines is essential for anyone working with triangles and trigonometry, as it provides a reliable method for solving complex geometric problems.

The General Formula and Its Variations

The general formula for the law of cosines is given as:

c2=a2+b2−2abcos⁡(C)c^2 = a^2 + b^2 - 2ab \cos(C)

Where:

  • a, b, and c are the lengths of the sides of the triangle.
  • C is the angle opposite side c.

This formula can be rearranged to solve for any side or angle in a triangle, making it a versatile tool in trigonometry. To solve for side a, the formula can be rearranged as:

a2=b2+c2−2bccos⁡(A)a^2 = b^2 + c^2 - 2bc \cos(A)

Similarly, to solve for side b, the formula becomes:

b2=a2+c2−2accos⁡(B)b^2 = a^2 + c^2 - 2ac \cos(B)

These variations of the law of cosines allow us to find the length of any side of a triangle if we know the lengths of the other two sides and the included angle. The angle included is the angle formed by the two known sides. For example, if we know sides a and b and the angle C, we can use the original formula to find side c. This flexibility is one of the key strengths of the law of cosines, making it applicable in a wide range of scenarios. Furthermore, the law of cosines can also be rearranged to solve for the angles of a triangle if we know the lengths of all three sides. By isolating the cosine term, we can find the cosine of the angle and then use the inverse cosine function to find the angle itself. This is particularly useful when we have a triangle where all three sides are known, and we need to determine the angles. Understanding these variations and how to apply them is crucial for mastering the law of cosines and using it effectively in problem-solving. Whether you are finding a side length or an angle, the law of cosines provides a reliable method for solving triangles.

Applying the Law to the Given Problem

In this specific problem, we are given a scenario where we need to find the correct equation that uses the law of cosines to solve for the length s. We are provided with several options, and our task is to identify which one accurately represents the law of cosines applied to the given triangle. To do this, we need to carefully examine the given information, including the known side lengths and angles, and then match them to the appropriate form of the law of cosines. The options presented likely involve different arrangements of the formula, and it's crucial to ensure that the correct sides and angles are plugged into the equation. This involves identifying which side is opposite the given angle and ensuring that the terms are arranged correctly according to the law of cosines formula. For example, if we are given two sides and the included angle, we need to use the form of the law that solves for the side opposite the given angle. If we are given all three sides, we might need to rearrange the formula to solve for the cosine of an angle. The process of selecting the correct equation involves a systematic approach of comparing each option to the general form of the law of cosines and verifying that the values are substituted correctly. This includes checking the coefficients, the trigonometric function, and the arrangement of the terms. By carefully analyzing each option and comparing it to the law of cosines, we can determine the equation that correctly represents the relationship between the sides and angles in the given triangle and accurately solves for the unknown side length s.

Analyzing the Options

Let's break down the given options and see which one correctly applies the law of cosines:

  • Option A: 92=s2+102−2(s)(10)cos⁡(100∘)9^2 = s^2 + 10^2 - 2(s)(10) \cos(100^{\circ})
  • Option B: 9=s+10−2(s)(10)cos⁡(100∘)9 = s + 10 - 2(s)(10) \cos(100^{\circ})
  • Option C: 102=s2+100−2(s)(10)cos⁡(100∘)10^2 = s^2 + 100 - 2(s)(10) \cos(100^{\circ})

To determine the correct equation, we need to understand how the law of cosines is applied when solving for a side length. The law of cosines states that for any triangle with sides a, b, and c, and angle C opposite side c:

c2=a2+b2−2abcos⁡(C)c^2 = a^2 + b^2 - 2ab \cos(C)

Comparing Option A to the general form of the law of cosines, we can see that it closely resembles the correct structure. The equation has the form of a side squared on one side, equal to the sum of the squares of the other two sides, minus twice the product of those sides and the cosine of the included angle. The values 9, s, and 10 seem to represent the side lengths, and 100 degrees is the angle opposite the side with length 9. Option B, on the other hand, is incorrect because it doesn't involve squaring the side lengths and doesn't follow the structure of the law of cosines. It appears to be a linear equation, which is not how the law of cosines is expressed. Option C also deviates from the correct form. While it has squared terms, the arrangement of the terms and the constants don't align with the law of cosines. Specifically, the term '100' on the right-hand side seems out of place, as it should be the square of a side length, and the equation doesn't clearly show the relationship between the sides and the included angle. Therefore, by carefully comparing the options to the law of cosines formula, we can identify the correct equation that accurately solves for the side length s.

The Correct Equation: Option A

Based on our analysis, Option A 92=s2+102−2(s)(10)cos⁡(100∘)9^2 = s^2 + 10^2 - 2(s)(10) \cos(100^{\circ}) is the equation that correctly uses the law of cosines to solve for the length s. This equation accurately represents the law of cosines formula, where the square of one side (9) is equal to the sum of the squares of the other two sides (s and 10) minus twice the product of those sides and the cosine of the included angle (100 degrees). The structure of the equation aligns perfectly with the standard form of the law of cosines, ensuring that the relationship between the sides and angles in the triangle is correctly represented. Option A also demonstrates a clear understanding of how to apply the law of cosines in a practical problem. The correct identification of the sides and the included angle is crucial for setting up the equation accurately. The equation also reflects the appropriate use of trigonometric functions and algebraic operations to solve for the unknown side length. In contrast, the other options either deviate from the correct formula or misrepresent the relationship between the sides and angles. This highlights the importance of understanding the underlying principles of the law of cosines and being able to apply it correctly in different scenarios. By choosing Option A, we demonstrate a solid grasp of trigonometric concepts and the ability to solve geometric problems using the law of cosines. Therefore, Option A is the most accurate and appropriate choice for solving for the length s in the given triangle.

Conclusion

In conclusion, understanding and applying the law of cosines is crucial for solving triangles when you don't have a right angle. By carefully analyzing the given options and comparing them to the general formula, we determined that Option A correctly uses the law of cosines to solve for the length s. This exercise highlights the importance of knowing the formula and how to apply it in different situations. The law of cosines is a powerful tool in trigonometry, and mastering it will significantly enhance your problem-solving abilities in geometry and related fields.