Express Sin(x + 7π/4) In Terms Of Sin(x) And Cos(x)

Oct 17, 2025 by Dimemap Team 52 views

Hey guys! Let's dive into a trigonometric problem where we'll rewrite $\sin(x + \frac{7\pi}{4})$ using $\sin(x)$ and $\cos(x)$ . This type of problem is a classic in trigonometry, and mastering it will definitely boost your understanding of trigonometric identities and angle sum formulas. So, grab your calculators (just kidding, you won't need them!), and let's get started!

Understanding the Angle Sum Identity

Before we jump into the specifics, let's quickly recap the angle sum identity for sine. This identity is the key to solving our problem. The angle sum identity states that:

$\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$

This formula allows us to break down the sine of a sum of two angles into a combination of sines and cosines of the individual angles. In our case, we have $a = x$ and $b = \frac{7\pi}{4}$ . So, we'll be using this identity to expand $\sin(x + \frac{7\pi}{4})$ . Remembering this identity is super important for tackling trig problems, so make sure you've got it down! We're going to use this identity to rewrite our expression, making it much easier to work with.

Now, let's dive deep into how this identity works and how we can apply it to our specific problem. The beauty of this formula is that it transforms a complex trigonometric function into simpler terms that we can easily evaluate. By understanding each component of the formula, we can confidently apply it to various trigonometric expressions. So, let’s break it down step by step to see how it all fits together.

Breaking Down the Identity

The angle sum identity, $\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$ , might look a bit intimidating at first, but it's actually quite straightforward once you understand its components. Think of it as a recipe where you mix different ingredients ( $\sin(a)$ , $\cos(b)$ , $\cos(a)$ , and $\sin(b)$ ) in a specific way to get your final dish ( $\sin(a + b)$ ). The key is to identify each ingredient correctly and then combine them according to the formula.

Let’s dissect each part:

$\sin(a + b)$ : This is the sine of the sum of two angles, $a$ and $b$ . It's the expression we want to simplify.
$\sin(a)$ : This is the sine of the first angle, $a$ .
$\cos(b)$ : This is the cosine of the second angle, $b$ .
$\cos(a)$ : This is the cosine of the first angle, $a$ .
$\sin(b)$ : This is the sine of the second angle, $b$ .

Why This Identity Works

The angle sum identity isn't just a random formula; it's derived from the fundamental geometry of triangles and the unit circle. Imagine two angles, $a$ and $b$ , on the unit circle. The sine and cosine of these angles correspond to the y- and x-coordinates, respectively, of the points where the angles intersect the circle. When you add these angles together, the sine of the resulting angle can be expressed in terms of the sines and cosines of the original angles, leading to the identity.

Common Mistakes to Avoid

One common mistake is to assume that $\sin(a + b)$ is simply $\sin(a) + \sin(b)$ . This is incorrect! The angle sum identity shows us that there's an additional term involving the cosines of the angles. Another mistake is mixing up the sine and cosine terms in the formula. Always remember the order: $\sin(a)\cos(b) + \cos(a)\sin(b)$ .

Understanding the angle sum identity is more than just memorizing a formula; it's about grasping the underlying principles that govern trigonometric functions. With a solid understanding, you can confidently tackle a wide range of trigonometric problems. So, let's move on and apply this knowledge to our specific problem: expressing $\sin(x + \frac{7\pi}{4})$ in terms of $\sin(x)$ and $\cos(x)$ .

Applying the Identity to Our Problem

Now that we've got the angle sum identity fresh in our minds, let's apply it to our problem: $\sin(x + \frac{7\pi}{4})$ . Remember, the identity is:

$\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$

In our case, $a = x$ and $b = \frac{7\pi}{4}$ . So, let's substitute these values into the identity:

$\sin(x + \frac{7\pi}{4}) = \sin(x)\cos(\frac{7\pi}{4}) + \cos(x)\sin(\frac{7\pi}{4})$

See how we've simply replaced $a$ with $x$ and $b$ with $\frac{7\pi}{4}$ ? This is the first crucial step. Now, we need to figure out the values of $\cos(\frac{7\pi}{4})$ and $\sin(\frac{7\pi}{4})$ . This is where our knowledge of the unit circle comes in handy!

Evaluating $\cos(\frac{7\pi}{4})$ and $\sin(\frac{7\pi}{4})$

The angle $\frac{7\pi}{4}$ might seem a bit intimidating, but it's actually a common angle on the unit circle. To visualize it, remember that $2\pi$ represents a full circle. $\frac{7\pi}{4}$ is just a bit less than $2\pi$ , meaning it's in the fourth quadrant. In fact, it's $\frac{\pi}{4}$ (45 degrees) short of a full circle. Thinking about it in terms of rotations helps a lot!

In the fourth quadrant, cosine is positive, and sine is negative. The reference angle for $\frac{7\pi}{4}$ is $\frac{\pi}{4}$ , which has well-known sine and cosine values. Specifically:

$\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$
$\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$

Since $\frac{7\pi}{4}$ is in the fourth quadrant, we have:

$\cos(\frac{7\pi}{4}) = \frac{\sqrt{2}}{2}$ (cosine is positive)
$\sin(\frac{7\pi}{4}) = -\frac{\sqrt{2}}{2}$ (sine is negative)

These values are super important, so make sure you're comfortable finding them on the unit circle. Now that we have these values, we can plug them back into our equation and simplify.

Back to the Equation

Let's substitute the values of $\cos(\frac{7\pi}{4})$ and $\sin(\frac{7\pi}{4})$ back into our expanded expression:

$\sin(x + \frac{7\pi}{4}) = \sin(x)\cos(\frac{7\pi}{4}) + \cos(x)\sin(\frac{7\pi}{4})$

$\sin(x + \frac{7\pi}{4}) = \sin(x)(\frac{\sqrt{2}}{2}) + \cos(x)(-\frac{\sqrt{2}}{2})$

Now, it's just a matter of simplifying. We can rewrite this as:

$\sin(x + \frac{7\pi}{4}) = \frac{\sqrt{2}}{2}\sin(x) - \frac{\sqrt{2}}{2}\cos(x)$

And that's it! We've successfully expressed $\sin(x + \frac{7\pi}{4})$ in terms of $\sin(x)$ and $\cos(x)$ . High five!

The Final Result and Its Significance

So, after all that trigonometric maneuvering, our final result is:

$\sin(x + \frac{7\pi}{4}) = \frac{\sqrt{2}}{2}\sin(x) - \frac{\sqrt{2}}{2}\cos(x)$

This equation tells us that the sine of the sum of $x$ and $\frac{7\pi}{4}$ can be written as a linear combination of $\sin(x)$ and $\cos(x)$ . This is a pretty cool result, and it highlights the power of trigonometric identities in simplifying complex expressions. But what does this actually mean, and why is it important?

Understanding the Transformation

What we've essentially done is taken a sine function with a phase shift (the $\frac{7\pi}{4}$ part) and rewritten it in a form that separates the sine and cosine components. This form can be incredibly useful in various applications, such as signal processing, physics, and engineering. For example, in signal processing, you might encounter signals that are phase-shifted, and rewriting them in terms of sine and cosine can make it easier to analyze their frequency content. This type of transformation really helps in understanding the underlying components of a wave.

Why This Matters

The ability to rewrite trigonometric expressions like this is crucial for several reasons:

Simplification: Sometimes, complex trigonometric expressions can be simplified into more manageable forms using identities. Our example is a perfect illustration of this.
Problem Solving: Many problems in calculus, physics, and engineering involve trigonometric functions. Being able to manipulate these functions is essential for solving these problems.
Understanding Functions: Rewriting functions in different forms can give us a deeper understanding of their behavior. In our case, we see how the phase shift affects the sine and cosine components.
Applications: As mentioned earlier, these transformations have practical applications in fields like signal processing, where understanding the composition of signals is crucial.

Practice Makes Perfect

Now that we've walked through this problem together, the best way to solidify your understanding is to practice similar problems. Try rewriting other trigonometric expressions using angle sum and difference identities. Experiment with different angles and see how the identities work in various scenarios. The more you practice, the more comfortable you'll become with these concepts.

Conclusion

So there you have it! We've successfully rewritten $\sin(x + \frac{7\pi}{4})$ in terms of $\sin(x)$ and $\cos(x)$ using the angle sum identity and a bit of unit circle knowledge. Remember, the key to mastering trigonometry is understanding the fundamental identities and practicing their applications. Don't be afraid to tackle complex problems – break them down into smaller steps, and you'll be surprised at what you can achieve. Keep practicing, and you'll become a trig whiz in no time! Keep exploring those trigonometric functions, guys, and you'll be amazed at how powerful these tools can be. Happy calculating! Remember, each problem you solve is a step closer to mastering trigonometry. So, keep challenging yourself and keep learning!