Simplifying Trig Expressions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of trigonometry and tackling a problem that might seem a bit intimidating at first glance: finding the exact value of the expression $\sin \frac{\pi}{4} \cos \frac{7 \pi}{12}+\cos \frac{\pi}{4} \sin \frac{7 \pi}{12}$. Don't worry, guys, it's not as scary as it looks. We'll break it down step by step, using some cool trigonometric identities to make the whole process super easy. By the end of this guide, you'll be able to simplify this expression and others like it with confidence. So, let's get started!

Unveiling the Trigonometric Identity

Okay, so the first thing we need to do is recognize the pattern. Does the expression $\sin \frac{\pi}{4} \cos \frac{7 \pi}{12}+\cos \frac{\pi}{4} \sin \frac{7 \pi}{12}$ look familiar? If you've spent some time with trig, you might see a resemblance to the angle sum identity for sine. If you are not familiar with it, don't worry, we'll quickly bring you up to speed! This identity states:

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

Notice how our given expression perfectly matches the right side of this identity? The angle $\frac{\pi}{4}$ can be thought of as A, and $\frac{7 \pi}{12}$ can be thought of as B. This is a crucial observation, because it tells us we can simplify the expression by reversing the identity, effectively turning a sum of products into a single sine function. Recognizing this pattern is like finding the secret code to unlock the solution. Once we know the code, we're golden. The entire expression transforms, making the simplification process much easier. It's like having a superpower that helps you solve complex problems. By recognizing the angle sum identity, we pave the way for a more streamlined approach to solving this trigonometric problem. Now, that we've identified the appropriate identity, let's apply it! It is one of the most fundamental identities in trigonometry, and it's super useful for simplifying expressions involving sines and cosines of sums or differences of angles. Understanding and remembering this identity will be very helpful for your mathematical journey.

Now, let's translate this into our equation. Essentially, we are working backward, and we can rewrite the initial expression as $\sin(\frac{\pi}{4} + \frac{7\pi}{12})$. Now the equation is much more simplified, because we are working with just one trigonometric function with a sum of angles.

Adding the Fractions: The Key Step

Alright, now that we've identified the identity, the next step is to simplify the angles inside the sine function. We have $\sin(\frac{\pi}{4} + \frac{7\pi}{12})$, so we need to add the two fractions inside the parenthesis. When adding fractions, we need a common denominator, right? Well, the least common denominator (LCD) for 4 and 12 is 12. So, we'll rewrite $\frac{\pi}{4}$ with a denominator of 12. To do this, we multiply both the numerator and the denominator by 3: $\frac{\pi}{4} * \frac{3}{3} = \frac{3\pi}{12}$. Now we can add the fractions:

$\frac{3\pi}{12} + \frac{7\pi}{12} = \frac{10\pi}{12}$

So our expression becomes $\sin(\frac{10\pi}{12})$.

Before we move on, let's take a closer look at the new fraction that we just created, $\frac{10\pi}{12}$. This fraction can be simplified, because both the numerator and denominator share a common factor of 2. Dividing both by 2, we get $\frac{10\pi}{12} = \frac{5\pi}{6}$. Therefore, the expression is now $\sin(\frac{5\pi}{6})$. Simplification of fractions is always a great practice to make sure you are always working with the easiest form of any problem. It is also an important step to ensure that the answer we finally obtain is also simplified.

Evaluating the Sine Function

Now we're at the final step! We need to evaluate $\sin(\frac{5\pi}{6})$. There are a couple of ways we can approach this. One is to recall the unit circle, or the knowledge of our special angles and trigonometric functions. Another is to use reference angles. Let's explore the unit circle method. Remember, the unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. The angle $\frac{5\pi}{6}$ is in the second quadrant. The sine function gives us the y-coordinate of the point on the unit circle that corresponds to the angle. The angle $\frac{5\pi}{6}$ is $\frac{\pi}{6}$ radians away from $\pi$. Therefore, the reference angle is $\frac{\pi}{6}$. We know that $\sin(\frac{\pi}{6}) = \frac{1}{2}$. Since sine is positive in the second quadrant, $\sin(\frac{5\pi}{6})$ is also positive. Therefore, $\sin(\frac{5\pi}{6}) = \frac{1}{2}$.

So, the exact value of the expression $\sin \frac{\pi}{4} \cos \frac{7 \pi}{12}+\cos \frac{\pi}{4} \sin \frac{7 \pi}{12}$ is $\frac{1}{2}$.

Breaking Down the Process: Step-by-Step Summary

Here's a quick recap, guys, of the steps we followed to simplify the expression:

1. Recognize the Identity: We started by identifying the angle sum identity for sine.
2. Apply the Identity: We used the identity to rewrite the expression as $\sin(\frac{\pi}{4} + \frac{7\pi}{12})$.
3. Add the Fractions: We added the fractions inside the sine function by finding a common denominator.
4. Simplify: We simplified the fraction and reduced the final expression to $\sin(\frac{5\pi}{6})$.
5. Evaluate: We evaluated the sine function using our knowledge of the unit circle, resulting in $\frac{1}{2}$.

Tips for Mastering Trig Identities

Want to become a trig ninja? Here are a few tips to help you master these kinds of problems:

• Memorize Key Identities: Start by memorizing the fundamental trig identities, such as the angle sum and difference identities for sine, cosine, and tangent. Having these at your fingertips will save you tons of time and effort.
• Practice, Practice, Practice: The more you practice, the better you'll get at recognizing patterns and applying the identities. Work through a variety of problems to build your confidence.
• Use the Unit Circle: Familiarize yourself with the unit circle. It's a powerful tool for visualizing angles and their corresponding sine, cosine, and tangent values.
• Break Down Complex Problems: Don't be afraid to break down complex expressions into smaller, more manageable steps. This will help you avoid making mistakes and keep track of your progress.

Conclusion: You Got This!

And there you have it! We've successfully simplified a seemingly complex trigonometric expression, breaking it down into manageable steps. Remember, the key is to understand the underlying identities and practice applying them. Trigonometry can be super fun when you grasp the fundamentals. Keep practicing, keep exploring, and you'll be amazed at what you can achieve. If you found this guide helpful, make sure to share it with your friends and classmates, and check out our other guides for more math tips and tricks. Happy calculating!