Finding The Exact Value Of Cos(195 Degrees) A Step-by-Step Guide
#h1
In the realm of trigonometry, finding the exact values of trigonometric functions for various angles is a fundamental skill. While calculators can provide decimal approximations, determining the precise value often requires a deeper understanding of trigonometric identities and angle relationships. In this comprehensive exploration, we will delve into the process of finding the exact value of cos(195°), unraveling the steps and concepts involved. This article aims to not only provide the solution but also to illuminate the underlying principles, making it an invaluable resource for students, educators, and anyone passionate about the intricacies of mathematics. Our journey will take us through the application of sum and difference formulas, the strategic use of known trigonometric values, and the simplification of radical expressions. By the end of this detailed analysis, you will have a clear understanding of how to approach similar problems and a strengthened appreciation for the elegance of trigonometric solutions. Let’s embark on this mathematical quest to decode the precise value of cos(195°).
Understanding the Problem: Breaking Down cos(195°)
#h2
To begin our exploration of finding the exact value of cos(195°), it's crucial to first understand why this problem requires a strategic approach. The angle 195° is not a standard angle found on the unit circle (such as 0°, 30°, 45°, 60°, 90°, etc.), for which we readily know the cosine values. Therefore, we cannot simply recall the value from memory or a basic trigonometric table. Instead, we need to decompose 195° into a sum or difference of angles whose cosine and sine values we do know. This is where our knowledge of trigonometric identities becomes essential. By expressing 195° as a combination of familiar angles, we can leverage trigonometric identities such as the sum and difference formulas for cosine to find the exact value. This process highlights the importance of understanding how angles relate to each other and how trigonometric functions behave under these relationships. In the following sections, we will explore how to strategically break down 195° and apply the appropriate trigonometric identities to arrive at the solution. The ability to decompose angles and apply trigonometric identities is a fundamental skill in trigonometry and is crucial for solving a wide range of problems. This section sets the stage for the detailed calculations that will follow, providing a clear understanding of the problem and the approach we will take to solve it. The challenge lies in identifying the right combination of angles that simplify the calculation while adhering to trigonometric principles.
Strategic Decomposition: Expressing 195° as a Sum of Known Angles
#h2
The key to finding the exact value of cos(195°) lies in expressing 195° as a sum or difference of angles for which we know the trigonometric values. Common angles that we are familiar with include 30°, 45°, 60°, and their multiples. Our goal is to find a combination of these angles that add up to 195°. A strategic approach involves considering different combinations and checking if they fit the bill. One possible decomposition is expressing 195° as the sum of 150° and 45°. Both 150° and 45° are angles whose cosine and sine values we can easily determine using the unit circle or special right triangles. Alternatively, we can express 195° as the sum of 135° and 60°, or even as the sum of 225° and -30°. However, the decomposition into 150° and 45° is a particularly convenient choice because both angles are within the first two quadrants, where we have a good understanding of the signs of trigonometric functions. This strategic decomposition is a crucial step in solving the problem, as it allows us to apply the sum formula for cosine. Without this decomposition, we would be stuck trying to find the cosine of an angle that doesn't have a readily available value. The ability to strategically decompose angles is a testament to a strong understanding of angle relationships and their representation on the unit circle. In the next section, we will apply the sum formula for cosine using this decomposition to find the exact value of cos(195°). This step-by-step approach ensures clarity and accuracy in our calculations.
Applying the Sum Formula for Cosine: cos(A + B)
#h2
Now that we have strategically decomposed 195° into the sum of 150° and 45°, we can apply the sum formula for cosine. This formula is a fundamental trigonometric identity that allows us to express the cosine of a sum of two angles in terms of the cosines and sines of the individual angles. The sum formula for cosine is given by: cos(A + B) = cos(A)cos(B) - sin(A)sin(B). In our case, A = 150° and B = 45°. To apply this formula, we need to find the values of cos(150°), cos(45°), sin(150°), and sin(45°). These values can be readily obtained from the unit circle or from our knowledge of special right triangles. cos(150°) is -√3/2, cos(45°) is √2/2, sin(150°) is 1/2, and sin(45°) is √2/2. Substituting these values into the sum formula, we get: cos(195°) = cos(150° + 45°) = cos(150°)cos(45°) - sin(150°)sin(45°) = (-√3/2)(√2/2) - (1/2)(√2/2). This application of the sum formula is a crucial step in solving the problem. It demonstrates the power of trigonometric identities in simplifying complex expressions. Without this formula, we would not be able to directly relate the cosine of 195° to the cosines and sines of familiar angles. In the next section, we will simplify this expression to arrive at the exact value of cos(195°). The careful application of the sum formula, combined with the correct values for the trigonometric functions of the individual angles, is essential for obtaining the correct result. This step highlights the interconnectedness of trigonometric concepts and the importance of mastering fundamental identities.
Calculation and Simplification: Finding the Exact Value
#h2
With the sum formula for cosine applied, we now have the expression for cos(195°) as: cos(195°) = (-√3/2)(√2/2) - (1/2)(√2/2). The next step is to simplify this expression to obtain the exact value. First, we multiply the fractions: (-√3/2)(√2/2) = -√6/4 and (1/2)(√2/2) = √2/4. Now, we substitute these values back into the expression: cos(195°) = -√6/4 - √2/4. Since both terms have a common denominator of 4, we can combine them into a single fraction: cos(195°) = (-√6 - √2)/4. This is the simplified form of the exact value of cos(195°). The simplification process involves basic arithmetic operations and an understanding of how to combine fractions with radicals. It's a crucial step in arriving at the final answer, as it presents the value in its most concise and easily interpretable form. The ability to simplify expressions involving radicals and fractions is a fundamental skill in mathematics, and this step reinforces the importance of mastering these techniques. The final result, cos(195°) = (-√6 - √2)/4, is a precise value that cannot be obtained directly from a calculator's decimal approximation. It showcases the power of trigonometric identities and strategic problem-solving in finding exact values. In the next section, we will verify this result and discuss its implications.
Verification and Conclusion: Confirming the Solution
#h2
After performing the calculations and simplifications, we have arrived at the exact value of cos(195°) as (-√6 - √2)/4. To ensure the accuracy of our solution, it is important to verify this result. One way to verify is to use a calculator to find the decimal approximation of both cos(195°) and (-√6 - √2)/4. If the decimal approximations match, then our solution is likely correct. Another method of verification involves understanding the quadrant in which 195° lies. Since 195° is between 180° and 270°, it lies in the third quadrant, where cosine values are negative. Our result, (-√6 - √2)/4, is indeed negative, which aligns with this understanding. Furthermore, we can compare our result with the given options to see if it matches any of them. In this case, our result matches option C, which is (-√6 - √2)/4. This verification step is crucial in the problem-solving process. It provides confidence in the solution and helps identify any potential errors in the calculations. The ability to verify results using different methods is a hallmark of a strong problem solver. In conclusion, by strategically decomposing 195° into the sum of 150° and 45°, applying the sum formula for cosine, simplifying the expression, and verifying the result, we have successfully found the exact value of cos(195°). This detailed exploration demonstrates the power of trigonometric identities and the importance of a systematic approach to solving mathematical problems. The process not only provides the answer but also enhances our understanding of trigonometric functions and their properties. This comprehensive analysis serves as a valuable resource for anyone seeking to master trigonometric concepts and problem-solving techniques.
By following these steps, we have not only found the exact value of cos(195°) but also gained a deeper understanding of the underlying trigonometric principles. This approach can be applied to similar problems, making it a valuable tool in your mathematical journey.
