Solving For C Evaluate And Simplify C² = 3² + 10² - 2(3)(10)cos(34°)
In the realm of mathematics, particularly trigonometry and geometry, solving for unknown variables within equations is a fundamental skill. This article delves into the process of solving for 'c' in the equation c² = 3² + 10² - 2(3)(10)cos(34°), a classic application of the law of cosines. We will meticulously evaluate and simplify the equation to arrive at the solution, rounding the final answer to the nearest tenth. This problem is a great illustration of how the law of cosines can be used to find the side of a triangle when you know two sides and the included angle. By understanding the steps involved, you'll be better equipped to tackle similar problems in various mathematical contexts. Our main keywords for this section are solve for c, law of cosines, and evaluate and simplify. This foundational knowledge is crucial for students and professionals alike in fields ranging from engineering to physics.
The Law of Cosines is a cornerstone of trigonometry, providing a relationship between the sides and angles of a triangle. Specifically, it states that for any triangle with sides a, b, and c, and angle C opposite side c, the following equation holds true: c² = a² + b² - 2abcos(C). This law is particularly useful when dealing with non-right triangles, where the Pythagorean theorem alone is insufficient. It's a generalization of the Pythagorean theorem, which applies only to right triangles. When angle C is 90 degrees, cos(C) becomes 0, and the law of cosines simplifies to the Pythagorean theorem: c² = a² + b². Understanding the nuances of the law of cosines is essential for solving various geometric problems, including those encountered in surveying, navigation, and physics. The ability to manipulate this equation and apply it correctly is a key skill in many mathematical and scientific disciplines. The equation provided in the problem, c² = 3² + 10² - 2(3)(10)cos(34°), is a direct application of the law of cosines, where a = 3, b = 10, and C = 34°. By carefully following the steps of evaluation and simplification, we can accurately solve for c and gain a deeper understanding of the underlying principles. The application of the law of cosines allows us to connect the lengths of the sides of a triangle to the cosine of one of its angles, making it a powerful tool for solving a wide range of problems.
To solve for c in the equation c² = 3² + 10² - 2(3)(10)cos(34°), we will break down the solution into manageable steps. First, we need to evaluate the individual terms in the equation. This involves squaring the numbers and calculating the cosine of the angle. The equation combines basic arithmetic operations with trigonometric functions, making it a comprehensive exercise in mathematical problem-solving. Each step must be performed with precision to ensure an accurate final result. Miscalculations in any step can lead to a significantly different answer. Secondly, we will simplify the equation by combining like terms and performing the necessary multiplications and subtractions. This simplification process is crucial for isolating the variable c and ultimately finding its value. Careful attention to the order of operations is paramount during this phase. Lastly, we will take the square root of both sides of the equation to solve for c, rounding the final answer to the nearest tenth as instructed. This final step completes the solution process and provides the numerical value of c. The entire process demonstrates the systematic approach required for solving mathematical problems involving trigonometric functions and algebraic manipulations. The steps involve applying the law of cosines, performing arithmetic operations, and using trigonometric functions, making it a thorough exercise in mathematical skills.
Step 1: Evaluate the Terms
In this initial step, we focus on evaluating each term in the equation c² = 3² + 10² - 2(3)(10)cos(34°). This involves calculating the squares of the numbers and finding the cosine of 34 degrees. First, let's address the squares: 3² equals 9, and 10² equals 100. These are straightforward arithmetic calculations. Next, we need to find the value of cos(34°). Using a calculator, we find that cos(34°) is approximately 0.829. It's crucial to use a calculator that is set to degrees mode, as the result will be different if it's set to radians. Now we have all the individual components needed for the next step. Evaluating these terms accurately is essential for obtaining the correct final answer. The equation now looks like this: c² = 9 + 100 - 2(3)(10)(0.829). The next step involves simplifying this expression by performing the multiplications and subtractions. This methodical approach ensures that we do not overlook any part of the equation. Evaluating the terms separately allows us to manage the complexity of the equation and reduce the likelihood of errors. This step lays the groundwork for the subsequent simplification and solution of the equation. The process of evaluating the terms involves both basic arithmetic and the use of trigonometric functions, highlighting the interconnectedness of different mathematical concepts. This step demonstrates the importance of precision and attention to detail in mathematical calculations.
Step 2: Simplify the Equation
After evaluating the individual terms, the next step is to simplify the equation c² = 9 + 100 - 2(3)(10)cos(34°). We've already determined that 3² = 9, 10² = 100, and cos(34°) ≈ 0.829. Substituting these values, the equation becomes c² = 9 + 100 - 2(3)(10)(0.829). Now, we need to perform the multiplication operations. First, 2(3)(10) equals 60. Then, we multiply 60 by 0.829, which gives us approximately 49.74. So, the equation now looks like this: c² = 9 + 100 - 49.74. Next, we perform the addition and subtraction operations from left to right. 9 + 100 equals 109. Then, we subtract 49.74 from 109, which gives us approximately 59.26. Therefore, the simplified equation is c² ≈ 59.26. This simplification process is crucial for isolating c² and preparing for the final step of taking the square root. The order of operations (PEMDAS/BODMAS) is critical here: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction. Adhering to this order ensures that we arrive at the correct simplified equation. By simplifying the equation step-by-step, we reduce the complexity and make it easier to solve for c. This methodical approach minimizes the chances of making errors in the calculations. The simplified equation c² ≈ 59.26 represents a significant milestone in the solution process, bringing us closer to the final answer. This step highlights the importance of careful arithmetic and adherence to the order of operations in mathematical problem-solving.
Step 3: Solve for c
Having simplified the equation to c² ≈ 59.26, the final step is to solve for c. To do this, we need to take the square root of both sides of the equation. The square root of c² is simply c. Now, we need to find the square root of 59.26. Using a calculator, we find that the square root of 59.26 is approximately 7.698. However, the instructions specify that we need to round the answer to the nearest tenth. Rounding 7.698 to the nearest tenth gives us 7.7. Therefore, the value of c is approximately 7.7. This final step completes the solution process, providing the numerical value of c that satisfies the given equation. Taking the square root is the inverse operation of squaring, and it allows us to isolate the variable c and find its value. Rounding the answer to the nearest tenth ensures that we adhere to the specified level of precision. The solution c ≈ 7.7 represents the length of the side c in the triangle described by the law of cosines equation. This step demonstrates the importance of understanding inverse operations and the proper use of rounding rules in mathematical calculations. The ability to solve for c in this manner is a fundamental skill in trigonometry and geometry. By following the steps of evaluation, simplification, and taking the square root, we have successfully found the value of c. This process underscores the importance of a systematic approach to solving mathematical problems.
After meticulously following the steps of evaluating, simplifying, and taking the square root, we have successfully solve for c in the equation c² = 3² + 10² - 2(3)(10)cos(34°). We found that c is approximately 7.7, rounded to the nearest tenth. This solution demonstrates the practical application of the law of cosines in determining the side length of a triangle when given two sides and the included angle. The final answer, 7.7, represents the value of c that satisfies the given equation. This result is crucial for various applications, including surveying, navigation, and engineering. The process of arriving at this answer involved careful arithmetic, trigonometric calculations, and algebraic manipulations. Each step was performed with precision to ensure the accuracy of the final result. The solution highlights the importance of a systematic approach to solving mathematical problems, breaking them down into manageable steps. The ability to solve for c in this context is a fundamental skill in mathematics and its related fields. This exercise provides a clear example of how the law of cosines can be used to evaluate and simplify complex equations and arrive at meaningful solutions. The final answer, c ≈ 7.7, is the culmination of the entire solution process, representing the successful application of mathematical principles to a specific problem. The entire process has reinforced the value of understanding the underlying concepts, following a logical sequence of steps, and paying attention to detail in mathematical calculations. This skill will be applicable to countless problems involving the law of cosines in particular and trigonometry in general.
In conclusion, we have successfully solve for c in the equation c² = 3² + 10² - 2(3)(10)cos(34°) by meticulously evaluating and simplifying the equation. Our step-by-step approach involved evaluating individual terms, simplifying the equation through arithmetic operations, and finally, taking the square root to isolate c. The final answer, rounded to the nearest tenth, is approximately 7.7. This exercise provides a clear illustration of the application of the law of cosines in a practical mathematical problem. The law of cosines is a fundamental concept in trigonometry and geometry, allowing us to relate the sides and angles of a triangle. Mastering this concept is crucial for solving various problems in fields such as engineering, physics, and surveying. The process of solving this equation also highlights the importance of precision and attention to detail in mathematical calculations. Each step, from evaluating the cosine of the angle to taking the square root, requires careful execution to ensure an accurate final answer. The ability to break down a complex problem into smaller, manageable steps is a valuable skill in mathematics and beyond. By following a systematic approach, we can effectively tackle challenging equations and arrive at the correct solutions. The successful solution of this problem demonstrates the power of mathematical principles and their application in real-world scenarios. The understanding and application of the law of cosines are essential tools for anyone working with triangles and geometric relationships. The experience of solve for c, evaluating and simplifying this equation reinforces the importance of a solid foundation in mathematical concepts and techniques.
