Exact Value Of Cos Pi 12

Finding the Exact Value of cos(π/12): A Comprehensive Guide

Determining the exact value of trigonometric functions like cos(π/12) often involves leveraging known values and trigonometric identities. This seemingly simple problem unveils a pathway to deeper understanding of fundamental trigonometric concepts and their applications. This article provides a detailed exploration of various methods to calculate cos(π/12), explaining the underlying principles and demonstrating multiple approaches for verification.

Understanding the Problem: cos(π/12)

The angle π/12 radians is equivalent to 15 degrees. While some angles like 30°, 45°, and 60° have readily available cosine values, π/12 requires a bit more work. We'll explore several methods to derive its exact value. Understanding the exact value is crucial not only for mathematical accuracy but also for applications in physics, engineering, and computer graphics, where precise calculations are essential.

Method 1: Using the Half-Angle Formula

This is arguably the most straightforward method. The half-angle formula for cosine is:

cos(x/2) = ±√[(1 + cos(x))/2]

We can cleverly use this formula by recognizing that π/12 is half of π/6 (30°). We know that cos(π/6) = √3/2. Substituting this into the half-angle formula:

cos(π/12) = cos(π/6 / 2) = ±√[(1 + cos(π/6))/2] = ±√[(1 + √3/2)/2]

To determine the sign, we consider the quadrant. π/12 is in the first quadrant, where cosine is positive. Therefore:

cos(π/12) = √[(1 + √3/2)/2] = √[(2 + √3)/4] = (√(2 + √3))/2

This is the exact value, but we can simplify it further using a technique shown in subsequent methods.

Method 2: Using the Difference Formula

We can express π/12 as a difference of two angles with known cosine values. Specifically:

π/12 = π/3 - π/4

The cosine difference formula is:

cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

Applying this to π/12:

cos(π/12) = cos(π/3 - π/4) = cos(π/3)cos(π/4) + sin(π/3)sin(π/4)

Substituting the known values:

cos(π/3) = 1/2 cos(π/4) = √2/2 sin(π/3) = √3/2 sin(π/4) = √2/2

cos(π/12) = (1/2)(√2/2) + (√3/2)(√2/2) = √2/4 + √6/4 = (√2 + √6)/4

This appears different from the result in Method 1, but we'll show that these are equivalent.

Method 3: Rationalizing the Denominator (Simplifying Method 1's Result)

Our result from Method 1 was (√(2 + √3))/2. This can be simplified by rationalizing the denominator. However, we need to rationalize the inner square root as well. This often involves a bit of algebraic manipulation. Let's consider the expression (√(2 + √3))/2 and aim to remove the nested radical. We can square it to simplify the process:

[(√(2 + √3))/2]² = (2 + √3)/4

This doesn't directly simplify our problem. Instead, let's look for an expression that, when squared, yields 2 + √3. This is often a trial-and-error process or can involve advanced techniques, but observing that (1 + √3/2)² = 1 + √3 + 3/4 = (7 + 4√3)/4 is helpful. However, this doesn't exactly match 2 + √3.

Let's try a different approach. We can multiply and divide by the conjugate of the numerator, which in this case isn't straightforward because it involves nested radicals. Therefore, the direct rationalization of the result from Method 1 isn't the easiest path.

Method 4: Verifying Equivalence of Results from Methods 1 and 2

We have two seemingly different expressions for cos(π/12):

• (√(2 + √3))/2 (from Method 1)
• (√2 + √6)/4 (from Method 2)

To show they are equivalent, we can square both expressions and demonstrate equality:

[(√(2 + √3))/2]² = (2 + √3)/4

[(√2 + √6)/4]² = (2 + 2√12 + 6)/16 = (8 + 4√3)/16 = (2 + √3)/4

Since the squares are equal, and both expressions are positive (as cos(π/12) is positive), the expressions themselves are equivalent. This verifies the consistency of our calculations.

Method 5: Using a Trigonometric Table or Calculator (Approximation)

While we're aiming for the exact value, a trigonometric table or calculator can provide an approximate decimal value. Using these tools, we find that cos(15°) ≈ 0.9659. This value serves as a useful check, but it's not the exact value, which we seek to express in radical form.

Conclusion: The Exact Value and its Significance

Through various methods, we have demonstrated that the exact value of cos(π/12) is:

(√2 + √6)/4 or equivalently (√(2 + √3))/2

While using a calculator provides a decimal approximation, expressing the value in exact radical form is often crucial for precise calculations in various fields. The process of finding the exact value reinforces our understanding of trigonometric identities, half-angle formulas, and the relationships between different angles. Furthermore, the exercises of verifying the equivalence of different expressions highlight the importance of algebraic manipulation and problem-solving skills. The ability to work with trigonometric functions and derive their exact values forms a foundational element in advanced mathematics and its practical applications. Remember that each method presented here offers a unique perspective on trigonometric operations and serves to strengthen your overall comprehension of this vital mathematical area.