What Is The Exact Value Of Cos(17π/8)?

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In the realm of trigonometry, determining the exact values of trigonometric functions for specific angles is a fundamental skill. One such intriguing problem involves finding the exact value of cos(17π/8). This seemingly complex expression can be simplified and solved using various trigonometric identities and principles. In this comprehensive guide, we will embark on a step-by-step journey to unravel the solution, exploring the underlying concepts and techniques involved. Our primary goal is to express cos(17π/8) in a more manageable form, allowing us to calculate its exact value. This exploration will not only enhance our understanding of trigonometry but also provide us with valuable problem-solving strategies applicable to a wide range of mathematical challenges. Before diving into the calculations, it's crucial to understand the properties of cosine and how angles relate to the unit circle. The cosine function represents the x-coordinate of a point on the unit circle corresponding to a given angle. Understanding this geometric interpretation is key to simplifying trigonometric expressions and finding exact values. Moreover, recognizing the periodicity of trigonometric functions is essential. Since the cosine function has a period of 2π, we can add or subtract multiples of 2π from the angle without changing its cosine value. This property will be instrumental in reducing the angle 17π/8 to a more manageable equivalent within the standard range of 0 to 2π. By combining these fundamental concepts and applying appropriate trigonometric identities, we will be able to precisely determine the exact value of cos(17π/8). This exercise will not only provide a concrete answer but also deepen our appreciation for the elegance and interconnectedness of trigonometric principles.

Breaking Down the Angle: Simplifying 17π/8

The first crucial step in solving for cos(17π/8) involves simplifying the angle itself. The angle 17π/8 is greater than 2π, making it necessary to find a coterminal angle within the range of 0 to 2π. We achieve this by subtracting multiples of 2π from 17π/8 until we arrive at an angle within this range. Subtracting 2π (which is equivalent to 16π/8) from 17π/8 gives us π/8. This means that 17π/8 and π/8 are coterminal angles, and therefore, cos(17π/8) = cos(π/8). This simplification significantly eases the problem, as we now need to find the cosine of a much smaller and more familiar angle. The significance of finding coterminal angles lies in the periodic nature of trigonometric functions. The cosine function, like sine, repeats its values every 2π radians. This property allows us to work with angles within a standard range, making calculations simpler and more intuitive. By reducing 17π/8 to π/8, we've essentially reframed the problem in terms of an angle whose trigonometric values are either known or can be easily derived. Now that we have simplified the angle, the next step involves employing trigonometric identities to further break down cos(π/8). Specifically, we will utilize the half-angle formula for cosine, which provides a direct relationship between the cosine of an angle and the cosine of half that angle. This formula will allow us to express cos(π/8) in terms of the cosine of a more readily calculable angle, setting the stage for our final solution. The ability to manipulate angles and apply trigonometric identities is a cornerstone of trigonometric problem-solving. This step-by-step approach, starting with angle simplification and progressing to identity application, exemplifies a powerful strategy for tackling complex trigonometric expressions.

Applying the Half-Angle Formula: A Key Trigonometric Identity

Now that we've simplified the angle to π/8, we can utilize the half-angle formula for cosine to find its exact value. The half-angle formula states that cos(x/2) = ±√((1 + cos(x))/2). In our case, x/2 = π/8, which means x = π/4. Therefore, we can rewrite our problem as finding cos(π/8) = cos((π/4)/2). Applying the half-angle formula, we get cos(π/8) = ±√((1 + cos(π/4))/2). The next step involves determining the sign (positive or negative) of the square root. Since π/8 lies in the first quadrant (0 < π/8 < π/2), where cosine is positive, we choose the positive square root. This is a critical step, as the correct sign ensures we arrive at the accurate value for the cosine. The decision to use the positive root stems directly from our understanding of the unit circle and the behavior of the cosine function in different quadrants. Now, we need to evaluate cos(π/4). From the special right triangles (45-45-90 triangle), we know that cos(π/4) = √2/2. Substituting this value into our equation, we get cos(π/8) = √((1 + √2/2)/2). This expression, while containing nested radicals, is a significant step closer to our final answer. We have successfully applied the half-angle formula and expressed cos(π/8) in terms of a known trigonometric value, cos(π/4). The application of the half-angle formula exemplifies the power of trigonometric identities in simplifying complex expressions. By strategically choosing and applying the appropriate identity, we can transform an unfamiliar problem into a more manageable one. This process highlights the importance of having a strong foundation in trigonometric identities and knowing when and how to use them.

Simplifying the Expression: Unveiling the Exact Value

Having applied the half-angle formula and obtained the expression cos(π/8) = √((1 + √2/2)/2), our next task is to simplify this expression to reveal the exact value. To begin, we simplify the fraction inside the square root: (1 + √2/2)/2 = (2 + √2)/4. Substituting this back into our equation, we get cos(π/8) = √( (2 + √2) / 4 ). Now, we can take the square root of the denominator, which is √4 = 2. This gives us cos(π/8) = √(2 + √2) / 2. This is the simplified exact value of cos(π/8). To match the format of the answer choices, we can rewrite the expression as cos(π/8) = √( (2 + √2) / 4 ). This form clearly shows the square root encompassing the entire fraction, aligning with the typical representation of such solutions. The process of simplifying the expression involves algebraic manipulation and a keen eye for detail. Each step, from combining fractions to taking square roots, requires careful execution to avoid errors. This meticulous approach is crucial in ensuring the accuracy of the final result. The final simplified expression, √( (2 + √2) / 4 ), represents the exact value of cos(17π/8). This value is an irrational number, highlighting the importance of trigonometric identities and algebraic techniques in finding precise solutions for trigonometric functions of non-standard angles. The journey from the initial problem to the final answer has showcased the power of combining trigonometric principles with algebraic manipulation. We started by simplifying the angle, then applied the half-angle formula, and finally simplified the resulting expression to arrive at the exact value. This methodical approach is a valuable strategy for tackling a wide range of mathematical problems.

Final Answer: Selecting the Correct Option

After simplifying the expression, we have determined that the exact value of cos(17π/8) is √( (2 + √2) / 4 ). Now, we need to compare this result with the given answer choices to select the correct option. The answer choices are:

  • A. √( (2 + √2) / 4 )
  • B. √( (2 - √2) / 4 )
  • C. 0.99
  • D. 0.38

By direct comparison, we can see that our calculated value matches option A: √( (2 + √2) / 4 ). This confirms that option A is the correct answer. The other options can be eliminated as follows:

  • Option B, √( (2 - √2) / 4 ), represents the value of sin(π/8), not cos(π/8).
  • Options C and D are decimal approximations. While they might be close to the actual value, they are not the exact value we were seeking. The problem specifically asked for the exact value, making approximations incorrect.

The process of selecting the correct option underscores the importance of understanding the nature of the solution. We were looking for an exact value, not an approximation, which immediately ruled out options C and D. Furthermore, recognizing the relationship between sine and cosine allowed us to quickly dismiss option B. This careful consideration of the answer choices is a crucial step in problem-solving, ensuring that we select the most accurate and appropriate solution. In conclusion, the exact value of cos(17π/8) is √( (2 + √2) / 4 ), and the correct answer is option A. This problem demonstrates the power of trigonometric identities and algebraic manipulation in finding precise solutions to trigonometric problems. The ability to simplify angles, apply appropriate identities, and simplify expressions is essential for success in trigonometry and related fields.

What is the exact value of cos(17π/8)?

Find the Exact Value of cos(17π/8) - A Step-by-Step Solution