Simplify Trigonometric Expression: Sin(a)cos(b) - Cos(a)sin(b)

by Andrew McMorgan 63 views

What's up, math lovers! Today, we're diving deep into the awesome world of trigonometry to tackle a super common problem. You've probably seen expressions like this before, and if you haven't, get ready, because they pop up everywhere in calculus, physics, and engineering. We're going to break down this specific problem: Which expression is equivalent to sin⁡(π12)cos⁡(7π12)−cos⁡(π12)sin⁡(7π12)?\sin \left(\frac{\pi}{12}\right) \cos \left(\frac{7 \pi}{12}\right)-\cos \left(\frac{\pi}{12}\right) \sin \left(\frac{7 \pi}{12}\right) ? And we'll see which of the given options, A. cos⁡(−π2)\cos \left(-\frac{\pi}{2}\right), B. sin⁡(−π2)\sin \left(-\frac{\pi}{2}\right), fits the bill. So, buckle up, grab your calculators (or your trusty unit circle!), and let's get this mathematical party started!

Unpacking the Trigonometric Identity

Alright guys, let's get straight to the heart of the matter. The expression sin⁡(π12)cos⁡(7π12)−cos⁡(π12)sin⁡(7π12)\sin \left(\frac{\pi}{12}\right) \cos \left(\frac{7 \pi}{12}\right)-\cos \left(\frac{\pi}{12}\right) \sin \left(\frac{7 \pi}{12}\right) looks a bit intimidating at first glance, right? But fear not! This is where our superhero trigonometric identities come into play. Specifically, this expression is a perfect example of the sine subtraction formula. Remember this classic? It states that for any angles AA and BB, we have:

sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\mathbf{\sin(A - B) = \sin A \cos B - \cos A \sin B}

Now, let's compare this identity to our problem. We can clearly see a direct match! If we let A=π12A = \frac{\pi}{12} and B=7π12B = \frac{7 \pi}{12}, our given expression sin⁡(π12)cos⁡(7π12)−cos⁡(π12)sin⁡(7π12)\sin \left(\frac{\pi}{12}\right) \cos \left(\frac{7 \pi}{12}\right)-\cos \left(\frac{\pi}{12}\right) \sin \left(\frac{7 \pi}{12}\right) is exactly in the form of sin⁡(A−B)\sin(A - B).

So, all we need to do is substitute our values for AA and BB into the formula. That gives us:

sin⁡(π12−7π12)\mathbf{\sin\left(\frac{\pi}{12} - \frac{7 \pi}{12}\right)}

Now, let's simplify the angle inside the sine function. Finding a common denominator, which is already 12, we get:

sin⁡(π−7π12)=sin⁡(−6π12)\mathbf{\sin\left(\frac{\pi - 7\pi}{12}\right) = \sin\left(\frac{-6\pi}{12}\right)}

And simplifying the fraction −6π12\frac{-6\pi}{12} gives us:

sin⁡(−π2)\mathbf{\sin\left(-\frac{\pi}{2}\right)}

Boom! We've simplified the original expression down to sin⁡(−π2)\sin\left(-\frac{\pi}{2}\right). This is exactly one of the options provided! So, option B is our winner. Isn't trigonometry neat? It's all about recognizing these patterns and applying the right tools. Keep practicing, and these identities will become second nature!

Evaluating the Simplified Expression and Options

So, we've brilliantly simplified the expression to sin⁡(−π2)\sin\left(-\frac{\pi}{2}\right). Now, let's take a moment to actually figure out what that value is and then check our options to be absolutely sure. The angle −π2-\frac{\pi}{2} radians is equivalent to -90 degrees. If you picture the unit circle, starting from the positive x-axis and moving clockwise by 90 degrees, you end up right on the negative y-axis. The coordinates of this point on the unit circle are (0,−1)(0, -1). Since the sine of an angle is represented by the y-coordinate on the unit circle, we know that:

sin⁡(−π2)=−1\mathbf{\sin\left(-\frac{\pi}{2}\right) = -1}

This is a fundamental value that's super useful to memorize. Now, let's look at the options given:

  • A. cos⁥(−π2)\cos \left(-\frac{\pi}{2}\right): The angle −π2-\frac{\pi}{2} radians is -90 degrees. On the unit circle, this corresponds to the point (0,−1)(0, -1). The cosine of an angle is the x-coordinate. Therefore, cos⁥(−π2)=0\cos\left(-\frac{\pi}{2}\right) = 0. This is not equal to -1, so option A is incorrect.

  • B. sin⁥(−π2)\sin \left(-\frac{\pi}{2}\right): As we just calculated, sin⁥(−π2)=−1\sin\left(-\frac{\pi}{2}\right) = -1. This exactly matches our simplified expression! So, this is our correct answer, guys!

  • C. Discussion category : mathematics: This isn't even a mathematical expression! It's just a label for the topic. So, clearly, C is not the answer.

See? By simplifying the original expression using the sine subtraction identity and then evaluating it, we've confirmed that sin⁡(−π2)\sin\left(-\frac{\pi}{2}\right) is indeed the equivalent expression. It's a good practice to not only find the equivalent expression but also know its numerical value, as it helps solidify your understanding and catches potential errors. Keep up the great work!

Why This Matters: The Power of Trigonometric Identities

Guys, understanding and being able to apply trigonometric identities like the sine subtraction formula is absolutely crucial in mathematics and its applications. It's not just about solving homework problems; it's about developing powerful tools for simplifying complex expressions and solving problems that would otherwise be incredibly difficult, if not impossible. Think about it: the original expression involves the product of sine and cosine of fractions of pi. Calculating the exact values of sin⁡(π12)\sin(\frac{\pi}{12}) and cos⁡(7π12)\cos(\frac{7\pi}{12}) directly would be a tedious process, likely involving sum or difference formulas for π12\frac{\pi}{12} (like π3−π4\frac{\pi}{3} - \frac{\pi}{4} or π4−π6\frac{\pi}{4} - \frac{\pi}{6}). But by recognizing the sin⁡(A−B)\sin(A-B) pattern, we bypassed all that complexity.

This ability to simplify is the bedrock of higher mathematics. In calculus, you'll encounter integrals and derivatives of trigonometric functions. Often, the first step to solving them is to use identities to rewrite the integrand or function in a simpler, more manageable form. For example, simplifying a complex trigonometric expression might transform it into a simple polynomial or exponential function that's easy to integrate. Without these identities, many calculus problems would remain unsolvable with standard techniques.

In physics, trigonometry is everywhere! From analyzing waves (sound waves, light waves, water waves) to understanding projectile motion, oscillations, and electrical circuits, trigonometric functions are the language used to describe periodic phenomena. Simplifying these descriptions using identities makes it easier to derive equations of motion, predict behavior, and understand the underlying physical principles. Imagine trying to model the swing of a pendulum or the vibration of a guitar string without the elegance of trigonometric simplification!

Engineering relies heavily on these concepts. Electrical engineers use Fourier series, which decompose complex signals into sums of simple sine and cosine waves, requiring a deep understanding of trigonometric manipulations. Mechanical engineers use trigonometry for structural analysis, robotics, and designing machinery. Even in computer graphics, transformations and rotations of objects in 2D and 3D space are handled using trigonometric functions and matrices.

So, the next time you see an expression that looks like sin⁡Acos⁡B−cos⁡Asin⁡B\sin A \cos B - \cos A \sin B, don't just see a bunch of symbols. See the sine subtraction identity waiting to simplify your life! It's a testament to the beauty and power of mathematical structure. Keep practicing, keep exploring, and you'll find that these identities are not just rules, but keys that unlock deeper understanding and problem-solving capabilities. You guys are doing great by engaging with these topics!

Conclusion: The Elegance of Equivalence

To wrap things up, we successfully tackled the problem of finding an equivalent expression for \sin \left(\frac{\pi}{12}\right) \cos \left(\frac{7 \pi}{12}\right)-\cos \left(\frac{\pi}{12} ight) \sin \left(\frac{7 \pi}{12} ight). By recognizing the sine subtraction identity, sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin(A - B) = \sin A \cos B - \cos A \sin B, we were able to substitute A=π12A = \frac{\pi}{12} and B=7π12B = \frac{7 \pi}{12}. This led us to the simplified form sin⁡(π12−7π12)\sin\left(\frac{\pi}{12} - \frac{7 \pi}{12}\right), which further reduced to sin⁡(−π2)\sin\left(-\frac{\pi}{2}\right). We then evaluated this expression and checked it against the given options. We found that option B, sin⁡(−π2)\sin\left(-\frac{\pi}{2}\right), was the exact equivalent expression. We also determined that \sin\left(-\frac{\pi}{2} ight) evaluates to -1, and confirmed that option A, cos⁡(−π2)\cos\left(-\frac{\pi}{2}\right), evaluates to 0. The category label was correctly identified as not being a mathematical expression.

This exercise beautifully demonstrates the elegance and power of trigonometric identities. They allow us to transform complex-looking expressions into simpler, more manageable forms, which is a fundamental skill in mathematics. It's like having a secret code that unlocks shortcuts to solutions. Keep practicing these, guys, and remember that every identity you master is another tool in your mathematical arsenal. Stay curious and keep exploring the fascinating world of math!