Finding S For Sec(s) = 1.5615 In The Interval [0, Pi/2]

July 27, 2025 by Scholario Team 56 views

Finding the Value of s in the Interval [0, π/2] for sec(s) = 1.5615

Hey everyone! Today, we're diving into a fun trigonometric problem where we need to find the value of s within a specific interval that satisfies a given equation. Specifically, we're looking for the value of s in the interval ${0, π/2}$ such that sec(s) = 1.5615. This might sound a bit intimidating at first, but don't worry, we'll break it down step by step and make it super clear. So, grab your calculators, and let's get started!

Understanding the Problem

Before we jump into solving the equation, let's make sure we understand what we're dealing with. The equation we have is sec(s) = 1.5615. Remember, secant (sec) is a trigonometric function that is the reciprocal of cosine (cos). In other words:

sec(s) = 1 / cos(s)

So, our equation can be rewritten as:

1 / cos(s) = 1.5615

The interval ${0, π/2}$ represents the first quadrant of the unit circle. This is important because trigonometric functions have different signs and behaviors in different quadrants. In the first quadrant, all trigonometric functions (sine, cosine, tangent, and their reciprocals) are positive. This simplifies our task a bit, as we don't need to worry about negative values in this case. Understanding these basics is super important, guys, because it sets the stage for solving the problem effectively. When we know the behavior of trig functions in different quadrants, it helps us narrow down the possible solutions and avoid common mistakes. It's like having a map before you start a journey – you know where you are and where you need to go. So, with this foundational knowledge, let’s move on to the next part where we actually start solving for s.

Solving for s

Now that we've got a handle on the problem, let's get down to the nitty-gritty and solve for s. We know that:

1 / cos(s) = 1.5615

To find cos(s), we can take the reciprocal of both sides:

cos(s) = 1 / 1.5615

Using a calculator, we find:

cos(s) ≈ 0.6404

Now we need to find the angle s whose cosine is approximately 0.6404. To do this, we use the inverse cosine function, also known as arccosine, denoted as cos⁻¹ or arccos. So:

s = arccos(0.6404)

Using a calculator (make sure it's in radian mode since our interval is in terms of π), we get:

s ≈ 0.8760 radians

This value is within our interval ${0, π/2}$ (since π/2 ≈ 1.5708), so it's a valid solution. It's always a good idea to check that your answer makes sense within the context of the problem. In this case, we were looking for an angle in the first quadrant, and 0.8760 radians falls squarely within that range. Plus, thinking about the cosine function, we know it starts at 1 at 0 radians and decreases to 0 at π/2 radians. So, a cosine value of 0.6404 makes sense for an angle somewhere in that first quadrant. Isn't it cool how all these concepts tie together? Next up, we'll round our answer to the required decimal places and wrap things up.

Rounding and Final Answer

Okay, we're almost there! We've found that s ≈ 0.8760 radians. The problem asks us to round our answer to four decimal places, which we've already done. So, our final answer is:

s ≈ 0.8760

Therefore, the value of s in the interval ${0, π/2}$ that satisfies sec(s) = 1.5615, rounded to four decimal places, is 0.8760. Woohoo! We solved it! Remember, the key to these problems is understanding the definitions of the trigonometric functions, knowing how to use inverse trigonometric functions, and paying attention to the given interval. Rounding correctly is also super important, guys, because you want to make sure you're giving the most accurate answer possible. Now, to make sure we've really nailed this concept, let’s do a quick recap of the steps we took. That way, you’ll have a clear roadmap for tackling similar problems in the future.

Recap of Steps

Let's quickly recap the steps we took to solve this problem. This will help solidify your understanding and give you a clear process to follow for similar questions:

Understand the Problem: We started by understanding that sec(s) is the reciprocal of cos(s) and that the interval ${0, π/2}$ represents the first quadrant.
Rewrite the Equation: We rewrote the equation sec(s) = 1.5615 as 1 / cos(s) = 1.5615.
Find cos(s): We found cos(s) by taking the reciprocal of 1.5615, which gave us cos(s) ≈ 0.6404.
Use the Inverse Cosine Function: We used the inverse cosine function (arccos) to find s: s = arccos(0.6404).
Calculate s: Using a calculator in radian mode, we found s ≈ 0.8760 radians.
Check the Interval: We verified that our solution, 0.8760, falls within the given interval ${0, π/2}$ .
Round the Answer: We rounded our answer to four decimal places, as required.

Following these steps makes the whole process much more manageable, right? Each step builds on the previous one, and before you know it, you've got the solution. Think of it like baking a cake – you follow the recipe step by step, and you end up with something delicious! These steps are your recipe for solving trig problems. So, keep practicing, and you'll become a pro in no time! In the next section, we'll touch on some common mistakes to avoid so you can ace these types of problems every time.

Common Mistakes to Avoid

To really master these types of problems, it's helpful to know the common pitfalls and how to avoid them. Here are a few mistakes that students often make when solving trigonometric equations, and how you can steer clear:

Forgetting the Reciprocal Relationship: A big mistake is forgetting that secant is the reciprocal of cosine. If you mix up the relationships between trigonometric functions, you'll end up going down the wrong path. Always remember: sec(s) = 1 / cos(s), csc(s) = 1 / sin(s), and cot(s) = 1 / tan(s). It’s like forgetting the key ingredient in a recipe – you just won’t get the right result.
Calculator Mode: Another very common error is having your calculator in the wrong mode (degrees vs. radians). Since our interval was given in terms of π, we needed to make sure our calculator was in radian mode. If you're in degree mode, you'll get a completely different answer. Always double-check your calculator mode before you start calculating! Think of it as checking your units of measurement – you wouldn’t measure a room in kilograms, would you?
Ignoring the Interval: It’s crucial to pay attention to the given interval. Trigonometric functions are periodic, which means they repeat their values. There might be multiple solutions to an equation, but only the one(s) within the specified interval are valid for that problem. Always make sure your solution falls within the given range. This is like staying within the boundaries of a game – if you go outside, you’re out!
Rounding Errors: Rounding too early or incorrectly can lead to an inaccurate final answer. Make sure to keep as many decimal places as possible during your calculations and only round at the very end. And always double-check that you're rounding to the correct number of decimal places, as specified in the problem. Rounding errors can be sneaky, guys, so be extra careful!

By keeping these common mistakes in mind, you'll be well-equipped to tackle trigonometric problems with confidence and accuracy. It's all about paying attention to the details and understanding the underlying concepts. Now, let's wrap up with a final thought.

Final Thoughts

So, there you have it! We've successfully found the value of s in the interval ${0, π/2}$ that satisfies sec(s) = 1.5615. We've broken down the problem, solved it step by step, and even talked about common mistakes to avoid. Remember, the key to mastering trigonometry (and any math topic, really) is practice, patience, and a solid understanding of the fundamentals. Don't be afraid to ask questions, work through examples, and take it one step at a time. You've got this! Trigonometry might seem daunting at first, but with each problem you solve, you'll build confidence and deepen your understanding. Keep exploring, keep learning, and keep having fun with math! You're all awesome, and I know you can conquer any trigonometric challenge that comes your way. Keep up the great work, guys!