Find The Exact Value Of Cos Pi/12

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

Determining the exact value of trigonometric functions for angles that aren't standard (like 30°, 45°, 60°) often requires employing trigonometric identities and a bit of clever manipulation. Finding the exact value of cos(π/12), or equivalently cos(15°), is a classic example of this. This article will walk you through several methods to achieve this, focusing on clarity and understanding, alongside highlighting key trigonometric concepts and strategies useful for similar problems.

Understanding the Problem: Cos(π/12)

Before diving into solutions, let's clarify the problem. We aim to express cos(π/12) without using a calculator, providing an exact value instead of a decimal approximation. Since π radians equals 180°, π/12 radians is equivalent to 15°. Therefore, we're essentially finding cos(15°).

Method 1: Using the Difference Formula for Cosine

This is arguably the most straightforward approach. We can express 15° as the difference between two angles with known cosine values: 45° and 30°. Specifically, 45° - 30° = 15°.

The difference formula for cosine states:

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

Let A = 45° and B = 30°. Then:

cos(15°) = cos(45° - 30°) = cos(45°)cos(30°) + sin(45°)sin(30°)

We know the exact values:

• cos(45°) = √2/2
• cos(30°) = √3/2
• sin(45°) = √2/2
• sin(30°) = 1/2

Substituting these values:

cos(15°) = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2) / 4

Therefore, the exact value of cos(π/12) is (√6 + √2) / 4.

Method 2: Using the Half-Angle Formula for Cosine

The half-angle formula for cosine allows us to find the cosine of half an angle if we know the cosine of the full angle. The formula is:

cos(θ/2) = ±√[(1 + cos θ) / 2]

The sign (+ or -) depends on the quadrant in which θ/2 lies. Since 15° is in the first quadrant, the cosine is positive. We can use this formula with θ = 30°:

cos(30°/2) = cos(15°) = √[(1 + cos 30°) / 2]

Substituting cos(30°) = √3/2:

cos(15°) = √[(1 + √3/2) / 2] = √[(2 + √3) / 4] = √(2 + √3) / 2

This looks different from our previous result, but let's rationalize the denominator and simplify:

To show that √(2 + √3) / 2 = (√6 + √2) / 4, we can square both sides:

(√(2 + √3) / 2)² = (2 + √3) / 4

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

Since both sides are equal when squared, and both are positive (since we are in the first quadrant), the expressions are equivalent. Therefore, both methods yield the same result.

Method 3: Using the Sum-to-Product Formula (Less Common but Illustrative)

While less intuitive for this specific problem, the sum-to-product formulas offer an alternative approach. However, they require more manipulation. We can express cos(15°) using angles whose cosine and sine values are known. This method is less direct and involves more steps but demonstrates another valuable tool in your trigonometric arsenal. The relevant sum-to-product formula isn't directly applicable to get cos(15°) in one step. We would need to work with other angles and then utilize the identities to reach our target angle. This method's complexity makes it less practical for this specific problem compared to the previous two methods.

Verifying the Result

Using a calculator (set to radians), you can approximate cos(π/12) to confirm our exact value:

cos(π/12) ≈ 0.9659

Now, let's approximate our exact value:

(√6 + √2) / 4 ≈ (2.449 + 1.414) / 4 ≈ 3.863 / 4 ≈ 0.96575

The slight discrepancy is due to rounding errors in the calculator's approximations of √2 and √6. The close agreement validates our calculations.

Further Applications and Extensions

The techniques used above – difference formula, half-angle formula – are broadly applicable for finding exact trigonometric values for various angles. Understanding these identities is crucial for solving more complex trigonometric equations and simplifying expressions.

For example, you can use these methods to find the exact values of:

• sin(π/12): Use the identity sin²x + cos²x = 1, knowing the value of cos(π/12).
• tan(π/12): Use the identity tan x = sin x / cos x.
• Other angles: By expressing angles as sums or differences of known angles, you can extend these methods to find exact trigonometric values for a wider range of angles.

Practical Tips and Strategies

• Memorize key values: Knowing the exact values of trigonometric functions for 30°, 45°, and 60° is essential.
• Identify suitable identities: Choose the identity that best fits the problem. The difference or half-angle formulas are frequently useful.
• Simplify expressions: Always simplify your results to the simplest radical form.
• Check your work: Use a calculator to approximate your answer and verify its reasonableness.
• Practice regularly: The more you practice, the more proficient you'll become in using trigonometric identities and solving these types of problems.

Conclusion

Finding the exact value of cos(π/12) showcases the power and elegance of trigonometric identities. While a calculator provides a decimal approximation, understanding the methods to derive the exact value enhances your mathematical understanding and problem-solving skills. The difference formula and half-angle formula, as demonstrated, provide efficient and reliable pathways to achieving this goal. Mastering these techniques opens the door to solving a wider range of trigonometric problems and strengthens your overall mathematical foundation. Remember to practice consistently, and you'll find these calculations become increasingly straightforward.