What Is The Exact Value Of Sin 112.5

What is the Exact Value of sin 112.5°? A Deep Dive into Trigonometric Identities

Determining the exact value of sin 112.5° might seem daunting at first glance, but with a systematic approach utilizing trigonometric identities and a touch of algebraic manipulation, we can unravel this mathematical puzzle. This article will not only provide the solution but also offer a detailed walkthrough, explaining the underlying principles and techniques. We'll explore multiple methods, ensuring a comprehensive understanding of this trigonometric problem.

Understanding the Angle: 112.5°

Before we dive into the calculations, it's crucial to understand the nature of the angle 112.5°. This angle lies in the second quadrant (90° < θ < 180°), where the sine function is positive. This knowledge will be useful in verifying our final result. Furthermore, 112.5° can be expressed as a half-angle, which is key to solving this problem using trigonometric half-angle formulas. Specifically, 112.5° = 225°/2. This representation will be instrumental in simplifying the calculations.

Method 1: Using the Half-Angle Formula

The most direct approach to finding the exact value of sin 112.5° involves the half-angle formula for sine:

sin(θ/2) = ±√[(1 - cos θ)/2]

Where the sign (+ or -) depends on the quadrant in which θ/2 lies. Since 112.5° is in the second quadrant, sin 112.5° is positive, so we'll use the positive square root. In our case, θ/2 = 112.5°, meaning θ = 225°.

Therefore, we have:

sin 112.5° = √[(1 - cos 225°)/2]

Now, we need to find the value of cos 225°. Since 225° = 180° + 45°, it lies in the third quadrant, where cosine is negative. We know that cos 45° = √2/2, so cos 225° = -√2/2.

Substituting this value into our equation:

sin 112.5° = √[(1 - (-√2/2))/2] = √[(1 + √2/2)/2] = √[(2 + √2)/4] = (√(2 + √2))/2

This is the exact value of sin 112.5°.

Method 2: Using the Sum-to-Product Formula (A More Elaborate Approach)

While the half-angle formula provides a concise solution, we can also explore a more elaborate method using the sum-to-product formulas. This method demonstrates a different approach and reinforces our understanding of trigonometric identities. We can express 112.5° as the sum of two known angles:

112.5° = 45° + 67.5°

However, this isn't immediately helpful. Let's try another combination:

112.5° = 135° - 22.5°

This also doesn't directly lend itself to readily available values. A more productive approach involves considering the double angle:

225° = 112.5° + 112.5°

We can use the double angle formula for sine:

sin(2θ) = 2sinθcosθ

Let θ = 112.5°. Then:

sin 225° = 2sin 112.5°cos 112.5°

We know that sin 225° = -√2/2. Substituting and rearranging:

-√2/2 = 2sin 112.5°cos 112.5° sin 112.5°cos 112.5° = -√2/4

This equation, unfortunately, introduces another unknown, cos 112.5°. While this approach is viable, it requires solving a system of equations involving both sine and cosine of 112.5°, which makes it less efficient than the half-angle method.

Method 3: Utilizing the Complementary Angle

We can also approach this by considering the complementary angle. The complementary angle to 112.5° is 180° - 112.5° = 67.5°. However, using the complementary angle identity (sin(90° - x) = cos x) doesn't directly lead to a simpler solution in this case. It requires further manipulation using half-angle formulas for cosine which ultimately leads to the same result as Method 1.

Verifying the Result: Why (√(2 + √2))/2 is Correct

We've established that sin 112.5° = (√(2 + √2))/2. Let's verify this result. As mentioned earlier, 112.5° lies in the second quadrant where sine is positive. Our calculated value is positive, which aligns with this expectation. Furthermore, we can approximate the numerical value:

(√(2 + √2))/2 ≈ 0.92388

Using a calculator to find sin 112.5°, we obtain approximately the same value, confirming the accuracy of our derived exact value.

Expanding on Trigonometric Identities: A Broader Perspective

This exercise highlights the importance of understanding various trigonometric identities and the ability to choose the most efficient approach based on the problem. The half-angle formula provided the most straightforward solution in this instance. However, exploring other methods, like the less efficient sum-to-product approach, enhances our understanding of the interconnectedness of these formulas and provides valuable practice in manipulating trigonometric expressions.

Beyond the Calculation: Applications and Implications

The ability to determine the exact value of trigonometric functions, even seemingly complex ones like sin 112.5°, is fundamental to various fields:

• Engineering and Physics: Solving problems involving oscillations, waves, and rotations frequently requires precise trigonometric calculations.
• Computer Graphics and Game Development: Accurate trigonometric calculations are essential for rendering realistic 3D graphics and simulating physical phenomena in games.
• Mapping and Navigation: Trigonometry plays a critical role in surveying, GPS technology, and various navigation systems.
• Mathematics and Higher-Level Studies: Understanding trigonometric identities is fundamental for advanced mathematical studies, including calculus, differential equations, and complex analysis.

Conclusion: Mastering Trigonometric Calculations

Finding the exact value of sin 112.5° is not just about obtaining a numerical result; it's about mastering fundamental trigonometric identities and applying them strategically. The half-angle formula proved to be the most efficient method in this case, providing a clear and concise path to the solution. However, exploring alternative approaches strengthens our understanding of the interrelationships between different trigonometric identities. Remember, the ability to manipulate trigonometric expressions is a crucial skill with far-reaching implications in various scientific and technological fields. By understanding the underlying principles and practicing these techniques, you'll build a strong foundation in trigonometry and its applications.