What Is The Exact Value Of Sin 105

What is the Exact Value of sin 105°? A Deep Dive into Trigonometric Identities

Determining the exact value of sin 105° might seem like a straightforward trigonometric problem, but it elegantly showcases the power and application of several key trigonometric identities. This isn't simply about plugging numbers into a calculator; it's about understanding the underlying mathematical principles and manipulating expressions to arrive at a simplified, exact answer. We'll explore several methods, highlighting the beauty and interconnectedness of trigonometry.

Understanding the Problem: Why Not a Calculator?

While a calculator readily provides an approximate decimal value for sin 105°, our goal is to find the exact value, expressed as a simplified radical. This precision is crucial in many mathematical and engineering applications where approximations can lead to significant errors. Finding the exact value involves employing trigonometric identities to break down sin 105° into more manageable components.

Method 1: Using the Sum-to-Product Formula

One effective approach is to use the sum-to-product formula for sine. We can express 105° as the sum of two angles whose sine and cosine values are readily known: 60° and 45°.

1. Express 105° as a sum:

105° = 60° + 45°

2. Apply the sum formula for sine:

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

Substituting A = 60° and B = 45°, we get:

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

3. Substitute known values:

We know that:

• 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. This is our final, simplified answer using this method.

Method 2: Utilizing the Difference Formula

Alternatively, we can express 105° as a difference of angles with known trigonometric values. For instance:

105° = 135° - 30° or 105° = 150° - 45°

Let's use 105° = 135° - 30° and apply the difference formula for sine:

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

sin(105°) = sin(135° - 30°) = sin 135° cos 30° - cos 135° sin 30°

We know that:

• sin 135° = √2/2
• cos 135° = -√2/2
• sin 30° = 1/2
• cos 30° = √3/2

Substituting these values:

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

Again, we arrive at the exact value of sin 105° as (√6 + √2) / 4. This confirms the result obtained using the sum formula.

Method 3: Employing the Half-Angle Formula

The half-angle formula provides another pathway to the solution. We can express 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 root:

sin(105°) = sin(210°/2) = √[(1 - cos 210°) / 2]

We know that cos 210° = -√3/2. Substituting this value:

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

This expression, while correct, isn't in the same simplified form as our previous results. To show equivalence, we need to rationalize the denominator and simplify further. This often involves algebraic manipulation and a deeper understanding of radical simplification. While it's a valid method, the sum/difference formulas are generally more direct for this specific problem.

Understanding the Significance of Exact Values

The seemingly small difference between an approximate and exact value can have significant consequences in various fields. For example:

• Engineering: In structural calculations, using an approximate value for sine could lead to errors in stress and strain calculations, potentially compromising the safety and stability of a structure.

• Physics: In wave phenomena or oscillatory motion, the precise value of trigonometric functions is crucial for accurate predictions and modeling.

• Computer Graphics: Accurate trigonometric calculations are essential for rendering realistic 3D graphics. Approximations can introduce distortions and artifacts.

Expanding on Trigonometric Identities: A Broader Perspective

Our exploration of sin 105° serves as a microcosm of the broader power and elegance of trigonometric identities. These identities aren't simply isolated formulas; they are interconnected tools that allow for the manipulation and simplification of trigonometric expressions. Mastering these identities opens up a world of problem-solving possibilities, allowing us to tackle seemingly complex trigonometric problems with grace and precision.

Further Exploration: Exploring Other Angles

The methods discussed above can be applied to find the exact values of sine (and other trigonometric functions) for a wide range of angles. By strategically choosing known angles and applying the appropriate sum, difference, or half-angle formulas, you can obtain exact values, avoiding the limitations and inaccuracies inherent in using calculator approximations.

Conclusion: The Beauty of Mathematical Precision

Finding the exact value of sin 105° is more than just an exercise in trigonometry; it’s a testament to the beauty and power of mathematical precision. The ability to manipulate trigonometric identities to arrive at a simplified, exact answer underscores the elegance and interconnectedness of mathematical concepts. This ability is essential not only for academic pursuits but also for various practical applications in science, engineering, and beyond. The journey to finding this exact value highlights the importance of understanding the underlying principles and selecting the most efficient approach to problem-solving. This understanding fosters a deeper appreciation for the rich tapestry of mathematics and its relevance to the real world.