Evaluating Trigonometric Expressions A Step-by-Step Guide To Cos[1/2 Cos^{-1}(cos(-14π/5))]

In this comprehensive exploration, we delve into the intricacies of trigonometric functions, specifically focusing on evaluating the expression cos[1/2 cos^-1(cos(-14π/5))]. This problem requires a solid understanding of inverse trigonometric functions, angle transformations, and the properties of cosine. By breaking down the expression step by step, we will arrive at the solution while reinforcing key concepts in trigonometry.

Breaking Down the Inner Cosine Function: cos(-14π/5)

Let’s start by analyzing the innermost part of the expression: cos(-14π/5). The goal here is to simplify the angle -14π/5 to an equivalent angle within the standard range of 0 to 2π or -π to π. This simplification will help us determine the cosine value more easily.

Angle Transformation

The angle -14π/5 is outside the typical range, so we need to find a coterminal angle within the range of 0 to 2π. We can do this by adding multiples of 2π to -14π/5 until we get an angle within this range. 2π is equivalent to 10π/5, so we can add multiples of 10π/5.

-14π/5 + 2(10π/5) = -14π/5 + 20π/5 = 6π/5

The angle 6π/5 falls within the range of 0 to 2π. Therefore, cos(-14π/5) = cos(6π/5). Now we need to determine the value of cos(6π/5).

Evaluating cos(6π/5)

The angle 6π/5 is in the third quadrant, where both cosine and sine are negative. We can express 6π/5 as π + π/5. Using the cosine addition formula, we have:

cos(6π/5) = cos(π + π/5) = cos(π)cos(π/5) - sin(π)sin(π/5)

Since cos(π) = -1 and sin(π) = 0, this simplifies to:

cos(6π/5) = -cos(π/5)

Thus, the inner part of the expression simplifies to -cos(π/5). This is a critical step in solving the entire problem.

Navigating the Inverse Cosine Function: cos^{-1}(-cos(π/5))

Now we move to the next layer of the expression: cos^-1(-cos(π/5)). The inverse cosine function, denoted as cos^-1(x) or arccos(x), gives the angle whose cosine is x. The range of the inverse cosine function is [0, π]. Therefore, we need to find an angle θ in this range such that cos(θ) = -cos(π/5).

Understanding the Negative Sign

The negative sign in front of cos(π/5) is crucial. We know that cos(π/5) is positive because π/5 is in the first quadrant (0 < π/5 < π/2). Therefore, -cos(π/5) is negative. We need to find an angle θ in the range [0, π] whose cosine is negative. This angle must lie in the second quadrant (π/2 < θ < π), where cosine values are negative.

Applying the Identity

We can use the identity cos(π - x) = -cos(x). In our case, x = π/5, so:

cos(π - π/5) = -cos(π/5)

This means that cos(4π/5) = -cos(π/5). The angle 4π/5 lies in the range [0, π], specifically in the second quadrant, which aligns with our earlier observation. Therefore:

cos^-1(-cos(π/5)) = 4π/5

This step significantly simplifies the expression, making it easier to handle the next operation.

Halving the Angle: 1/2 * cos^{-1}(-cos(π/5))

Next, we multiply the result from the previous step by 1/2: (1/2) * (4π/5) = 2π/5. So now we have 1/2 cos^-1(-cos(π/5)) = 2π/5. This reduces the complexity further, bringing us closer to the final answer. This step is straightforward but essential in the overall solution.

The Final Cosine Evaluation: cos(2π/5)

Finally, we need to evaluate the outermost cosine function: cos(2π/5). This is a standard trigonometric value that can be derived using various methods, including geometric arguments or trigonometric identities.

Determining cos(2π/5)

The angle 2π/5 is in the first quadrant, so its cosine value is positive. To find the exact value of cos(2π/5), we can use the double angle formula and some algebraic manipulation. Let θ = π/5. Then 2θ = 2π/5, and 3θ = 3π/5. We know that 5θ = π.

We can write 3θ = π - 2θ. Taking the cosine of both sides:

cos(3θ) = cos(π - 2θ)

Using the identity cos(π - x) = -cos(x), we get:

cos(3θ) = -cos(2θ)

Now, we can express cos(3θ) and cos(2θ) in terms of cos(θ) using trigonometric identities:

4cos³(θ) - 3cos(θ) = -(2cos²(θ) - 1)

Let x = cos(θ) = cos(π/5). The equation becomes:

4x³ - 3x = -2x² + 1

Rearranging the terms, we get:

4x³ + 2x² - 3x - 1 = 0

We know that cos(π) = -1 is a solution to this equation, so (x + 1) is a factor. We can perform polynomial division to find the other factors:

(4x³ + 2x² - 3x - 1) / (x + 1) = 4x² - 2x - 1

So, we have (x + 1)(4x² - 2x - 1) = 0. Since x = cos(π/5) ≠ -1, we solve the quadratic equation 4x² - 2x - 1 = 0 using the quadratic formula:

x = [2 ± √(4 + 16)] / 8 = [2 ± √20] / 8 = (1 ± √5) / 4

Since cos(π/5) is positive, we take the positive root:

cos(π/5) = (1 + √5) / 4

Now we use the double angle formula to find cos(2π/5):

cos(2π/5) = 2cos²(π/5) - 1

cos(2π/5) = 2([(1 + √5) / 4]²) - 1

cos(2π/5) = 2[(1 + 2√5 + 5) / 16] - 1

cos(2π/5) = (6 + 2√5) / 8 - 1

cos(2π/5) = (3 + √5) / 4 - 1

cos(2π/5) = (√5 - 1) / 4

However, we don't need the exact value here. We just need to express the final result, which is cos(2π/5). Therefore, the final value of the expression is cos(2π/5).

Conclusion

In summary, by carefully breaking down the expression cos[1/2 cos^-1(cos(-14π/5))] step by step, we have determined its value to be cos(2π/5). This problem highlights the importance of understanding trigonometric identities, inverse functions, and angle transformations. Each step, from simplifying the angle to applying the inverse cosine and finally evaluating the cosine, requires a solid foundation in trigonometry. This detailed solution not only answers the question but also serves as a valuable review of essential trigonometric principles.