Solve Cos S = -sqrt(3)/2: Exact Values Of S In [0, 2pi)

Hey there, math enthusiasts and curious minds! Ever found yourself staring at a trigonometric equation like cos s = -sqrt(3)/2 and wondered, "How on earth do I find those exact values of s?" Well, you're in the right place, because today we're going to break down this problem step-by-step, making it super clear and totally approachable. We're specifically looking for values of s within the interval [0, 2pi), which basically means one full rotation around the unit circle, starting from 0 and going up to, but not including, 2pi. This problem isn't just about getting the right answer; it's about understanding the fundamental concepts of trigonometry, the unit circle, and how to apply them. Mastering these skills will give you a rock-solid foundation for all your future math adventures, whether you're dealing with advanced calculus, physics, or even engineering. So, buckle up, grab your virtual pen and paper, and let's dive deep into the world of cosines, reference angles, and quadrants to uncover those elusive exact values. We'll explore everything from the basics of the cosine function to using the mighty unit circle as our secret weapon, ensuring that by the end of this article, you'll be confidently solving similar problems like a pro. This journey will not only give you the answers but also the why behind them, making you a true master of trigonometric problem-solving. Let's get started and unravel the mystery of cos s = -sqrt(3)/2 together, in a way that feels natural, friendly, and super engaging.

Understanding the Cosine Function: A Quick Refresher

Before we dive headfirst into solving cos s = -sqrt(3)/2, let's take a quick pit stop and make sure we're all on the same page about what the cosine function actually represents. Seriously, guys, this is the bedrock of our understanding! At its core, in a right-angled triangle, the cosine of an angle s is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. But when we're talking about finding exact values and working with angles beyond just acute ones (0 to 90 degrees), the unit circle becomes our ultimate guide. On the unit circle, which is a circle with a radius of 1 centered at the origin (0,0) of a coordinate plane, any point (x, y) on the circle's circumference corresponds to an angle s measured counterclockwise from the positive x-axis. Here's the cool part: the x-coordinate of that point (x, y) is precisely the value of cos s. So, when we're asked to find cos s = -sqrt(3)/2, we're essentially looking for the angles where the x-coordinate on the unit circle is exactly -sqrt(3)/2. The cosine function is also periodic, meaning its values repeat over certain intervals. Specifically, it has a period of 2pi (or 360 degrees), which is why our problem restricts us to the [0, 2pi) interval – we only need to find the solutions within one full cycle. Knowing the domain (all real numbers) and range (values between -1 and 1, inclusive) of the cosine function is also super helpful, as it quickly tells us if a solution is even possible. Since -sqrt(3)/2 is indeed between -1 and 1 (it's approximately -0.866), we know we're on the right track and solutions definitely exist! Getting a solid grip on these fundamentals isn't just academic; it empowers you to visualize these abstract concepts and makes solving problems much more intuitive. Trust me, spending a little extra time here pays huge dividends down the road. Understanding what cosine is conceptually prevents you from just memorizing formulas and instead builds a real, lasting comprehension. This foundation will be critical as we move to the unit circle, which will visually map out where these values actually live.

The Unit Circle: Your Best Friend for Exact Values

Alright, team, let's talk about the unit circle – seriously, if trigonometry were a superhero team, the unit circle would be Captain America. It's that foundational and reliable! As we briefly touched upon, the unit circle is a circle with a radius of one unit, centered at the origin of a Cartesian coordinate system. Why is it our best friend for finding exact values like cos s = -sqrt(3)/2? Because it visually represents all possible angles and their corresponding cosine and sine values (the x and y coordinates, respectively) in a super intuitive way. Imagine starting at the point (1, 0) on the positive x-axis; this represents an angle of 0 radians. As you move counterclockwise around the circle, you're sweeping through different angles, and for each angle, the x-coordinate gives you cos s and the y-coordinate gives you sin s. The unit circle is elegantly divided into four quadrants, and knowing which quadrant you're in tells you a lot about the sign (positive or negative) of your trigonometric functions. For instance, in Quadrant I (0 to pi/2), both x and y are positive, so cos s and sin s are positive. In Quadrant II (pi/2 to pi), x is negative and y is positive, meaning cos s is negative while sin s is positive. Quadrant III (pi to 3pi/2) has both x and y negative, so both cos s and sin s are negative. Finally, Quadrant IV (3pi/2 to 2pi) has x positive and y negative, so cos s is positive and sin s is negative. This is incredibly important for our problem, cos s = -sqrt(3)/2, because it immediately tells us that our solutions must lie in either Quadrant II or Quadrant III, where the x-coordinate (cosine value) is negative. Beyond just signs, the unit circle is also invaluable for memorizing special angles and their exact trigonometric values. Think about angles like pi/6 (30 degrees), pi/4 (45 degrees), and pi/3 (60 degrees). Their cosine and sine values, along with those for 0, pi/2, pi, and 3pi/2, form the backbone of many trigonometry problems. Knowing these key points on the unit circle allows you to quickly deduce values for related angles using symmetry. For example, if you know cos(pi/6) = sqrt(3)/2, then you can use that knowledge, combined with the quadrant rules, to find other angles with the same reference value but different signs. This systematic approach, leveraging the visual power of the unit circle, transforms what might seem like a complex problem into a straightforward exercise in pattern recognition and logical deduction. It’s truly the key to unlocking exact values without needing a calculator, and it builds a deep understanding of trigonometric relationships that will serve you well in all sorts of mathematical contexts. So, consider the unit circle your ultimate cheat sheet and visual aid – it’s a game-changer!

Tackling cos s = -sqrt(3)/2: The Step-by-Step Breakdown

Alright, enough with the warm-up! It's time to put our knowledge into action and actually solve cos s = -sqrt(3)/2. This is where all those concepts we just discussed come together. Remember, we're looking for exact values of s within the interval [0, 2pi). Don't feel overwhelmed, we're going to break this down into three manageable steps that will guide you straight to the solution. Think of it like following a recipe – if you stick to the steps, you'll get a delicious (and correct!) result every time. The key here is to not just jump to an answer but to systematically work through the process, building confidence with each stage. This methodical approach ensures accuracy and helps you understand why each angle is a solution, rather than just memorizing a list. We'll start by identifying a crucial piece of the puzzle: the reference angle, which is like the core building block for our solutions. Then, we'll use our understanding of the unit circle and quadrants to figure out where cosine is negative. Finally, we'll combine these two pieces to pinpoint the exact angles we're looking for within the specified interval. Ready to solve this thing like the math rockstars you are?

Step 1: Identify the Reference Angle (s_ref)

The very first thing we need to do when solving an equation like cos s = -sqrt(3)/2 is to find its reference angle. Think of the reference angle, s_ref, as the acute angle (meaning between 0 and pi/2 or 0 and 90 degrees) that has the same absolute value for its cosine. In our case, cos s = -sqrt(3)/2, so we need to find an angle s_ref such that cos(s_ref) = | -sqrt(3)/2 | = sqrt(3)/2. This is where your knowledge of special angles on the unit circle really comes in handy! If you've spent some time getting friendly with the unit circle, you'll immediately recall that cos(pi/6) equals sqrt(3)/2. (If you're wondering why, remember pi/6 corresponds to 30 degrees, and in a 30-60-90 triangle, the side adjacent to the 30-degree angle is sqrt(3) times half the hypotenuse, and on the unit circle, the x-coordinate for pi/6 is sqrt(3)/2). So, our reference angle s_ref is pi/6. This angle is absolutely critical because it acts as a template for finding all other angles that satisfy the condition. The reference angle itself might not be one of our final answers if the original cosine value is negative (which it is here!), but it's the building block we use to construct those answers. Without knowing this base angle, finding the actual solutions would be like trying to navigate without a map. Understanding how to quickly identify reference angles is a fundamental skill that streamlines the entire process of solving trigonometric equations. It simplifies the problem by allowing us to temporarily ignore the negative sign and focus on the core numerical value, which is usually associated with a well-known special angle. Once we have this s_ref, we can then use it in conjunction with our knowledge of quadrants to find the angles where the cosine is actually negative. So, pat yourself on the back for nailing this first crucial step: s_ref = pi/6. Now, let's move on to figuring out where this sqrt(3)/2 needs to be negative on our unit circle.

Step 2: Determine the Quadrants where Cosine is Negative

Okay, so we've got our reference angle, s_ref = pi/6. Now, we need to think about the original problem again: cos s = -sqrt(3)/2. Notice that negative sign! This tells us that the x-coordinate on the unit circle must be negative. If you remember our chat about the unit circle and its quadrants, this part becomes super easy. We know that the x-coordinate (which represents cos s) is positive in Quadrant I (0 to pi/2) and Quadrant IV (3pi/2 to 2pi). Conversely, the x-coordinate is negative in Quadrant II (pi/2 to pi) and Quadrant III (pi to 3pi/2). This means our solutions for s must lie in Quadrant II and Quadrant III. There's a handy mnemonic many students use to remember the signs in each quadrant: All Students Take Calculus (ASTC). It goes like this: In Quadrant I (All), all trigonometric functions are positive. In Quadrant II (Students), only Sine (and its reciprocal, cosecant) is positive. In Quadrant III (Take), only Tangent (and its reciprocal, cotangent) is positive. In Quadrant IV (Calculus), only Cosine (and its reciprocal, secant) is positive. Since we're looking for where cos s is negative, we skip Quadrant I and Quadrant IV. We're left with Quadrant II (where only sine is positive, so cosine is negative) and Quadrant III (where only tangent is positive, so cosine is negative). This step is crucial because it narrows down our search significantly. Instead of looking everywhere, we've identified exactly two regions on the unit circle where our angles can exist. Without this understanding of quadrant signs, you might accidentally include angles from Quadrant I or IV, leading to incorrect solutions. Being able to quickly pinpoint the correct quadrants based on the sign of the trigonometric value is a hallmark of a strong understanding of trigonometry. It's like having a treasure map that tells you exactly which islands to search for the hidden gold! So, armed with the knowledge that our angles must be in Quadrants II and III, we're perfectly set up for the final calculation step. Keep that s_ref = pi/6 in mind, because we're about to use it to construct our actual solutions in these specific quadrants. We're almost there, guys!

Step 3: Calculate the Angles in [0, 2pi)

Alright, moment of truth! We've identified our reference angle as s_ref = pi/6, and we know our solutions must reside in Quadrant II and Quadrant III because that's where cos s is negative. Now, let's put it all together to calculate the exact values of s within our specified interval [0, 2pi). This is where we use the reference angle to 'build' our solutions in the correct quadrants.

First, let's find the angle in Quadrant II. In Quadrant II, an angle is related to its reference angle by the formula pi - s_ref. This makes perfect sense if you visualize it on the unit circle: you go pi (halfway around), and then you 'backtrack' by the reference angle s_ref to land in Quadrant II. So, for our problem:

s_1 = pi - s_ref s_1 = pi - pi/6 To subtract these, we need a common denominator. pi is equivalent to 6pi/6. s_1 = 6pi/6 - pi/6 s_1 = 5pi/6

This 5pi/6 is our first solution! It falls perfectly within the [0, 2pi) interval, and if you think about 5pi/6 on the unit circle, its x-coordinate is indeed -sqrt(3)/2. Awesome!

Next, let's find the angle in Quadrant III. In Quadrant III, an angle is related to its reference angle by the formula pi + s_ref. Again, visualize it: you go pi (halfway around), and then you 'add' the reference angle s_ref to push you into Quadrant III. So, for our problem:

s_2 = pi + s_ref s_2 = pi + pi/6 Using our common denominator trick again: s_2 = 6pi/6 + pi/6 s_2 = 7pi/6

And 7pi/6 is our second solution! It also fits nicely within the [0, 2pi) interval, and its x-coordinate on the unit circle is indeed -sqrt(3)/2. Fantastic!

We've found both angles that satisfy the condition cos s = -sqrt(3)/2 within the specified interval [0, 2pi). The solutions are 5pi/6 and 7pi/6. It's always a good idea to quickly check your answers. For 5pi/6, we know it's in Quadrant II, so cosine should be negative. For 7pi/6, it's in Quadrant III, so cosine should also be negative. Both checks align with our original problem's negative value. This methodical calculation, combining the reference angle with quadrant knowledge, ensures you find all correct solutions without missing any or including extraneous ones. Mastering this final step is what brings the whole problem together, allowing you to confidently present your exact values.

Why These Values Matter and What's Next?

So, there you have it, guys! We've successfully navigated the trigonometric seas and found the exact values for s where cos s = -sqrt(3)/2 in the interval [0, 2pi). Our solutions are 5pi/6 and 7pi/6. But why does this specific type of problem matter beyond just passing your next math test? Well, understanding how to find these exact values is a cornerstone of so much more advanced mathematics and science. These aren't just abstract numbers; they represent fundamental relationships that govern everything from wave patterns in physics and engineering to signal processing and even computer graphics. When you analyze a periodic phenomenon, like sound waves or alternating current, you'll often encounter trigonometric equations, and knowing how to solve for precise angles is absolutely vital. Imagine designing an electronic circuit or modeling the path of a pendulum – these exact values help predict behavior with incredible accuracy. This skill also builds your mathematical intuition. Instead of just plugging numbers into a calculator, you're now visualizing the unit circle, understanding reference angles, and logically deducing solutions. This deeper engagement strengthens your problem-solving abilities across the board, making you a more versatile and confident thinker. Furthermore, while we restricted our answers to [0, 2pi), it's worth noting that due to the periodic nature of the cosine function, there are infinitely many solutions if we remove that restriction. You would simply add or subtract 2n*pi (where n is any integer) to our solutions. For example, 5pi/6 + 2pi, 5pi/6 - 2pi, 7pi/6 + 2pi, and so on, would all be valid solutions. However, for most practical applications and the scope of typical coursework, identifying the principal values within a single cycle like [0, 2pi) is usually the primary objective. So, what's next for you? My advice is to practice, practice, practice! Try solving similar problems with different cosine or sine values, both positive and negative. Work through other intervals. The more you engage with these types of problems, the more ingrained these concepts will become, and the faster and more confidently you'll be able to solve them. You've now got a powerful tool in your mathematical toolkit, and you're well on your way to mastering trigonometry. Keep exploring, keep questioning, and keep having fun with math!

In conclusion, by understanding the definition of the cosine function, leveraging the power of the unit circle, identifying reference angles, and determining appropriate quadrants, we systematically arrived at the exact values for s. The journey from a basic equation to precise answers is a testament to the elegant and interconnected nature of trigonometry. You've done a fantastic job breaking down this problem, and I'm confident you'll ace your next challenge!