Hey guys! Let's dive into evaluating a cool trigonometric expression: $\sin^2 135^{\circ} + \cos^2 135^{\circ}$. This might look a bit intimidating at first, but trust me, it's simpler than it seems. We're going to break it down step by step, making sure everyone understands the core concepts and how to apply them. Trigonometric expressions like this one pop up all the time in math, physics, and engineering, so getting comfortable with them is super important. We'll not only solve this specific problem but also explore the underlying principles that make it tick. By the end of this article, you'll be a pro at handling similar expressions and understand why they work the way they do.
Understanding the Basics: Sine and Cosine
Okay, first things first, let's quickly recap what sine and cosine actually represent. Imagine a unit circle – that's a circle with a radius of 1 centered at the origin of a coordinate plane. Now, picture an angle, let's call it , formed by the positive x-axis and a line that intersects the circle. The point where this line meets the circle has coordinates (x, y). Guess what? The cosine of (written as ${\cos \theta}$) is simply the x-coordinate of that point, and the sine of (written as ${\sin \theta}$) is the y-coordinate. This is fundamental to understanding trigonometric functions. For angles between 0° and 90°, sine and cosine are both positive. But what about angles larger than 90°, like our 135°? That's where things get a little more interesting, and we need to think about which quadrant the angle falls into. In the second quadrant (between 90° and 180°), sine is positive, and cosine is negative. This is because the y-coordinate is still positive, but the x-coordinate is now negative. So, as we move around the unit circle, the values of sine and cosine change signs depending on the quadrant. This is crucial for accurately evaluating trigonometric expressions for any angle, not just those between 0° and 90°. We can visualize these relationships using the unit circle, which is an incredibly useful tool for understanding trigonometric functions. The unit circle helps us to see how sine and cosine relate to each other and how their values change as the angle increases. By mastering the unit circle, you can quickly determine the sine and cosine of many common angles, like 0°, 30°, 45°, 60°, 90°, and their multiples. This understanding forms the basis for tackling more complex trigonometric problems.
Finding Sine and Cosine of 135°
Now, let's zoom in on our specific angle: 135°. To find the sine and cosine of 135°, we can use our knowledge of the unit circle and reference angles. A reference angle is the acute angle formed between the terminal side of the angle and the x-axis. For 135°, the reference angle is 180° - 135° = 45°. This is super handy because we know the sine and cosine values for 45°: ${\sin 45^{\circ} = \frac{\sqrt{2}}{2}}$ and ${\cos 45^{\circ} = \frac{\sqrt{2}}{2}}. Remember, 135° is in the second quadrant, where sine is positive, and cosine is negative. So, \${\sin 135^{\\circ}\} will have the same value as ${\sin 45^{\circ}}$ (positive), and ${\cos 135^{\circ}}$ will have the negative of ${\cos 45^{\circ}}. Therefore, \${\sin 135^{\\circ} = \\frac{\\sqrt{2}}{2}\} and ${\cos 135^{\circ} = -\frac{\sqrt{2}}{2}}$. Understanding reference angles allows us to relate angles in different quadrants back to angles in the first quadrant, where we typically know the sine and cosine values. This technique is essential for simplifying trigonometric calculations and accurately determining the values of trigonometric functions for any angle. By using reference angles, we can break down complex problems into simpler steps, making them easier to solve. This approach not only helps in finding the sine and cosine of 135° but also applies to a wide range of trigonometric problems involving different angles.
Evaluating the Expression: $\sin^2 135^{\circ} + \cos^2 135^{\circ}$
Alright, we've got $\sin 135^{\circ} = \frac{\sqrt{2}}{2}}$ and ${\cos 135^{\circ} = -\frac{\sqrt{2}}{2}}$. Now we can plug these values into our original expression + \cos^2 135^\circ}$. Squaring the sine and cosine values, we get}2}\right)^2 + \left(-\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} + \frac{2}{4}$. Simplifying this, we have{2} + \frac{1}{2} = 1$. So, the value of the expression $\sin^2 135^{\circ} + \cos^2 135^{\circ}$ is simply 1! Isn't that neat? This result might seem like a coincidence, but it's actually a manifestation of a fundamental trigonometric identity. This identity is a cornerstone of trigonometry and shows up in countless applications. It’s a powerful tool for simplifying expressions and solving equations. The beauty of this identity lies in its simplicity and universality; it holds true for any angle, not just 135°. By recognizing and applying this identity, you can significantly simplify trigonometric problems and gain a deeper understanding of trigonometric relationships. So, remember this result: it’s not just a solution to this specific problem, but a valuable piece of your trigonometric toolkit.
The Pythagorean Identity: A Key Takeaway
Now, here's the real kicker: what we just did perfectly illustrates a crucial trigonometric identity called the Pythagorean Identity. It states that for any angle , $\sin^2 \theta + \cos^2 \theta = 1$. This identity is a direct consequence of the Pythagorean theorem applied to the unit circle. Remember how sine and cosine are the y and x coordinates, respectively, of a point on the unit circle? The radius of the unit circle is 1, and the Pythagorean theorem tells us that x² + y² = 1². Substituting sine for y and cosine for x, we get our identity! This identity is incredibly important because it provides a fundamental relationship between sine and cosine. It allows us to express sine in terms of cosine, and vice versa, which is super useful for simplifying expressions and solving equations. The Pythagorean Identity is one of the most frequently used identities in trigonometry. It’s a powerful tool for simplifying complex trigonometric expressions and is essential for solving trigonometric equations. Mastering this identity will significantly enhance your ability to work with trigonometric functions and tackle more advanced problems. It's not just a formula to memorize; it's a core concept that underpins many trigonometric principles. Understanding its derivation from the Pythagorean theorem provides a deeper appreciation for its significance and applicability.
Why This Matters: Applications and Beyond
So, why bother with all this? Well, trigonometric expressions and identities are everywhere in STEM fields. They show up in physics when describing wave motion, in engineering when analyzing electrical circuits, and in computer graphics when rotating objects in 3D space. The Pythagorean Identity, in particular, is a workhorse in many of these applications. It allows us to simplify calculations, solve equations, and gain a deeper understanding of the relationships between different quantities. For example, in physics, the Pythagorean Identity is used to relate the horizontal and vertical components of velocity or force. In engineering, it helps in analyzing alternating current circuits. And in computer graphics, it's used to perform rotations and transformations without distortion. By mastering these fundamental trigonometric concepts, you're not just learning math for the sake of math; you're building a foundation for success in a wide range of fields. The ability to manipulate trigonometric expressions and apply identities is a valuable skill that will serve you well in your academic and professional pursuits. So, keep practicing, keep exploring, and keep discovering the power of trigonometry!
In conclusion, evaluating $\sin^2 135^{\circ} + \cos^2 135^{\circ}$ not only gives us a numerical answer (which is 1!) but also reinforces our understanding of sine, cosine, reference angles, and the crucial Pythagorean Identity. Keep up the great work, and you'll be trig pros in no time!