Decoding The Exact Value Of Sin(π/12) A Mathematical Exploration

Hey guys! Let's dive into a cool mathematical exploration today. We're going to tackle the value of sin(π/12) and figure out if the given expression, (√(2-√3))/2, is the exact value or just a close approximation. Buckle up, because we're about to break this down in a way that's super clear and easy to understand.

Is (√(2-√3))/2 the Real Deal for sin(π/12)?

When we talk about the value of sin(π/12), we're essentially asking for the sine of 15 degrees (since π/12 radians is equal to 15 degrees). Now, at first glance, the expression (√(2-√3))/2 might seem a bit intimidating, but don't worry, we're going to unravel it. To determine if this value is exact, we need to understand what “exact” means in the context of trigonometry and how these values are derived. In mathematics, an exact value is one that can be expressed without any rounding or approximation. This often means expressing trigonometric values in terms of surds (expressions involving square roots) rather than decimal approximations. The sine function, like other trigonometric functions, has exact values for certain angles such as 0, 30, 45, 60, and 90 degrees (and their multiples and related angles). These exact values are derived from the unit circle and special right triangles (30-60-90 and 45-45-90 triangles). For angles that aren't these standard ones, like 15 degrees (π/12 radians), we often use trigonometric identities to find their exact values.

So, the key question here is: can we derive (√(2-√3))/2 using fundamental trigonometric principles and arrive at it without any approximations? The answer, as we'll see, is a resounding yes! We can use trigonometric identities to break down sin(π/12) into expressions involving known, exact values. This involves using half-angle formulas or sum-and-difference formulas, which are powerful tools in our trigonometric toolkit. These formulas allow us to express the sine of an angle in terms of the sine and cosine of related angles, often leading us to expressions we can simplify to exact values. By applying these identities step-by-step, we will demonstrate how the given expression is indeed the exact value of sin(π/12).

Cracking the Code: How to Find the Exact Value

Okay, let's get our hands dirty and actually calculate the value of sin(π/12). There are a couple of ways we can approach this, but one of the most common is using the half-angle formula. This formula is a gem because it allows us to find the sine (or cosine, or tangent) of half an angle if we know the cosine of the full angle. The half-angle formula for sine is: sin(x/2) = ±√((1 - cos(x))/2). Remember guys, the ± sign means we need to consider the quadrant in which the angle x/2 lies to determine the correct sign.

In our case, we want to find sin(π/12), which is the same as sin(15°). We can think of 15° as half of 30°, which is a friendly angle whose sine and cosine we know exactly. So, let's set x/2 = π/12, which means x = π/6 (or 30°). We know that cos(π/6) = cos(30°) = √3/2. Plugging this into our half-angle formula, we get: sin(π/12) = sin(15°) = ±√((1 - √3/2)/2). Now, we need to simplify this expression. First, let's focus on the sign. Since 15° is in the first quadrant, where sine is positive, we can ditch the ± and keep only the positive root. This gives us: sin(π/12) = √((1 - √3/2)/2). Next, let's simplify the fraction inside the square root. We can rewrite 1 as 2/2, so we have: sin(π/12) = √(((2 - √3)/2)/2). Dividing by 2 is the same as multiplying by 1/2, so we get: sin(π/12) = √((2 - √3)/4). Now, here's a neat trick: we can take the square root of the numerator and the denominator separately: sin(π/12) = √(2 - √3) / √4. Since √4 = 2, we finally arrive at: sin(π/12) = (√(2 - √3))/2. Boom! That's exactly the expression we were given. This step-by-step derivation shows us that this value is not an approximation but the exact value of sin(π/12).

Another Path: The Sum-and-Difference Formula

There's more than one way to skin a cat, as they say! We can also find the exact value of sin(π/12) using the sum-and-difference formulas. These formulas allow us to express trigonometric functions of sums or differences of angles in terms of trigonometric functions of the individual angles. This method provides an alternative approach and reinforces our understanding of trigonometric identities.

The angle π/12 (15°) can be expressed as the difference between two angles whose trigonometric values we know exactly. A common choice is to write 15° as 45° - 30° (or π/4 - π/6 in radians). The sum-and-difference formula for sine is: sin(A - B) = sin(A)cos(B) - cos(A)sin(B). Let's apply this formula with A = π/4 (45°) and B = π/6 (30°). We know the following exact values: sin(π/4) = √2/2, cos(π/4) = √2/2, sin(π/6) = 1/2, cos(π/6) = √3/2. Plugging these values into our formula, we get: sin(π/12) = sin(π/4 - π/6) = (√2/2)(√3/2) - (√2/2)(1/2). Now, let's simplify: sin(π/12) = (√6/4) - (√2/4). We can combine these fractions since they have a common denominator: sin(π/12) = (√6 - √2)/4. This looks different from our previous result, (√(2 - √3))/2, but don't panic! They are actually equivalent. To see this, we would need to use a clever algebraic manipulation to show that (√(2 - √3))/2 is indeed equal to (√6 - √2)/4. This involves squaring both expressions and demonstrating that they result in the same value. While the algebraic proof is a bit involved, the important takeaway is that both methods lead to the exact same value for sin(π/12), just expressed in different forms. This further solidifies the fact that (√(2 - √3))/2 is not an approximation but the precise value of sin(π/12).

Why Exact Values Matter

Okay, so we've shown that (√(2-√3))/2 is the exact value of sin(π/12). But you might be wondering,