Find The Exact Value Of Sin 105

Finding the Exact Value of sin 105°: A Comprehensive Guide

Determining the exact value of trigonometric functions for angles beyond the standard 0°, 30°, 45°, 60°, and 90° often requires employing trigonometric identities and properties. This article provides a detailed walkthrough of calculating the exact value of sin 105°, exploring multiple approaches and highlighting the underlying mathematical principles. We'll delve into the use of sum and difference formulas, half-angle formulas, and even consider the connection to complex numbers for a more comprehensive understanding.

Understanding the Problem: Why Not Just Use a Calculator?

While a calculator readily provides an approximate value for sin 105°, approximately 0.9659, our goal here is to find the exact value – an expression that doesn't involve decimal approximations. This pursuit enhances our understanding of trigonometric relationships and demonstrates the power of mathematical manipulation.

Method 1: Sum and Difference Formulas

The most straightforward approach involves utilizing the sum formula for sine:

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

We can express 105° as the sum of two angles whose sine and cosine values are readily known:

105° = 60° + 45°

Substituting these values into the sum formula:

sin(105°) = sin(60° + 45°) = sin 60° cos 45° + cos 60° sin 45°

Now, we substitute the known values:

sin 60° = √3/2 cos 60° = 1/2 sin 45° = √2/2 cos 45° = √2/2

Therefore:

sin(105°) = (√3/2)(√2/2) + (1/2)(√2/2) = (√6 + √2) / 4

Therefore, the exact value of sin 105° is (√6 + √2) / 4

Method 2: Difference Formula for Sine

Alternatively, we can express 105° as a difference of known angles:

105° = 150° - 45°

Using the difference formula for sine:

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

sin(105°) = sin(150° - 45°) = sin 150° cos 45° - cos 150° sin 45°

We know that:

sin 150° = sin (180° - 30°) = sin 30° = 1/2 cos 150° = cos (180° - 30°) = -cos 30° = -√3/2 sin 45° = √2/2 cos 45° = √2/2

Substituting these values:

sin(105°) = (1/2)(√2/2) - (-√3/2)(√2/2) = (√2 + √6) / 4

Again, we arrive at the exact value of sin 105° as (√6 + √2) / 4

Method 3: Half-Angle Formula

The half-angle formula offers another approach. We can consider 105° as half of 210°. The half-angle formula for sine is:

sin(A/2) = ±√[(1 - cos A) / 2]

Here, A = 210°. Since 105° lies in the second quadrant where sine is positive, we use the positive square root:

cos 210° = cos (180° + 30°) = -cos 30° = -√3/2

sin(105°) = sin(210°/2) = √[(1 - (-√3/2)) / 2] = √[(2 + √3) / 4]

This expression, while exact, isn't in the simplest radical form. To simplify, we need to rationalize the denominator and manipulate the expression, which ultimately leads us back to (√6 + √2) / 4 (This step requires advanced algebraic manipulation and is beyond the scope of a concise explanation here, but the equivalence can be verified).

Method 4: Exploring Complex Numbers (Advanced)

A more advanced method involves using Euler's formula:

e^(ix) = cos x + i sin x

where 'i' is the imaginary unit (√-1).

We can write:

e^(i105°) = cos 105° + i sin 105°

This approach requires knowledge of complex numbers and De Moivre's theorem and is generally more complex than the previous methods for this specific problem. While it provides a powerful alternative for solving more intricate trigonometric problems, it's less efficient for calculating sin 105°.

Verifying the Result: Approximate Calculation

To verify our exact value, we can approximate (√6 + √2) / 4:

√6 ≈ 2.449 √2 ≈ 1.414

(2.449 + 1.414) / 4 ≈ 0.96575

This closely matches the approximate value obtained from a calculator (0.9659), confirming the accuracy of our exact value calculation.

Practical Applications and Significance

Understanding how to derive the exact values of trigonometric functions is crucial in various fields:

• Calculus: Exact values are essential for evaluating integrals and derivatives involving trigonometric functions. Approximations can introduce significant errors in these calculations.

• Physics and Engineering: Many physical phenomena, such as wave motion and oscillations, are modeled using trigonometric functions. Precise calculations are vital for accurate predictions and designs.

• Computer Graphics and Game Development: Accurate trigonometric calculations underpin the rendering of 3D graphics and the physics simulations in video games.

• Surveying and Navigation: Trigonometric functions are fundamental in calculating distances, angles, and positions.

Conclusion: Mastering Trigonometric Identities

This detailed explanation demonstrates multiple paths to finding the exact value of sin 105°. While a calculator offers a quick approximation, understanding the underlying mathematical principles and employing trigonometric identities provides a deeper understanding and enhances problem-solving skills. The ability to manipulate trigonometric expressions is essential for advanced mathematical and scientific endeavors. Remember that choosing the most efficient method often depends on the specific problem and the available tools. The sum and difference formulas generally offer the most straightforward approach for angles that can be easily expressed as sums or differences of known angles. The importance of mastering these identities cannot be overstated. They are the bedrock of more advanced trigonometric concepts and applications.