Unveiling Trigonometric Errors Correcting A Tricky Problem Step-by-Step

Hey guys! Today, we are diving into a super interesting trigonometry problem that involves finding the six trigonometric values for an angle ${\theta}$. This angle’s terminal ray passes through the point ${(-7, -24)}$, which puts it in a standard position. A student tackled this problem, but it looks like there might be some errors in their solution. Our mission? To dissect their work, pinpoint the mistakes, and set things straight. So, grab your calculators, and let’s get started!

The Trigonometric Puzzle A Step-by-Step Breakdown

Setting the Stage Understanding the Given Information

First off, let’s break down what we know. We have a point ${(-7, -24)}$ on the terminal ray of angle ${\theta}$. This is crucial because it tells us a lot about where our angle lies in the coordinate plane. Remember, the standard position means the angle starts at the positive x-axis and rotates. The coordinates ${(-7, -24)}$ tell us we are in the third quadrant, where both x and y values are negative. This is super important because the signs of our trigonometric functions depend on the quadrant. In the third quadrant, tangent and cotangent are positive, while sine, cosine, cosecant, and secant are negative. This fundamental concept is the cornerstone for solving this problem accurately. We need to keep this in mind as we evaluate the student's answers, ensuring that the signs align with the quadrant rules. Trigonometry can seem daunting, but with a systematic approach, we can unravel even the most complex problems. So, let's keep this quadrant analysis at the forefront as we move forward, ensuring our final answers not only have the correct numerical values but also the appropriate signs. This initial step of quadrant identification is a vital safeguard against common errors in trigonometric calculations, making our solution robust and reliable. The power of understanding quadrants cannot be overstated in trigonometry!

Calculating the Radius The Key to Unlocking Trigonometric Values

Next up, we need to find the distance from the origin to the point ${(-7, -24)}$. This distance, often called the radius ${r}$, is the hypotenuse of the right triangle formed by the point, the x-axis, and the origin. We can calculate ${r}$ using the Pythagorean theorem: ${r = \sqrt{x^2 + y^2}}$. Plugging in our values, we get ${r = \sqrt{(-7)^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25}$. So, ${r = 25}$. This value is essential because it forms the denominator in our trigonometric ratios. The radius, being a distance, is always positive, which simplifies our calculations. This step is a critical bridge connecting the coordinates of the point to the trigonometric values. Without the correct radius, the trigonometric ratios will be inaccurate. The Pythagorean theorem is our trusty tool here, providing a straightforward method to find this crucial length. This foundational calculation sets the stage for accurately determining sine, cosine, and all other trigonometric functions. A precise radius is the backbone of accurate trigonometric values, ensuring our calculations are rooted in solid geometric principles. So, let's make sure we've nailed this down before we proceed further!

Student's Answers and the Hunt for Discrepancies

Now, let's take a peek at the student’s answers. They have given us the six trigonometric values, and our task is to verify each one. This is where the detective work begins! We need to compare the student's results with what we know about the signs of the trigonometric functions in the third quadrant and the values we can calculate using our ${x}$, ${y}$, and ${r}$. This comparison is crucial for identifying errors. By systematically checking each trigonometric value, we can pinpoint exactly where the student went wrong. This process not only helps us correct the solution but also deepens our understanding of trigonometric principles. It's like a puzzle where each piece (trigonometric value) must fit perfectly within the framework of quadrant rules and ratio definitions. A careful examination of the student's work allows us to catch mistakes and reinforce our knowledge. This step is not just about finding errors; it's about enhancing our trigonometric intuition and problem-solving skills. So, let's put on our detective hats and get ready to scrutinize each value, ensuring accuracy and alignment with trigonometric truths!

The student's answers are:

• ${\sin(\theta) = \frac{24}{25}}$
• ${\csc(\theta) = \frac{25}{24}}$
• ${\cos(\theta) = -\frac{7}{25}}$
• ${\sec(\theta) = -\frac{25}{7}}$
• ${\tan(\theta) = \frac{24}{7}}$
• ${\cot(\theta) = \frac{7}{24}}$

Spotting the Errors A Deep Dive into Trigonometric Functions

Sine and Cosecant Unveiling the Sign Error

The student stated that ${\sin(\theta) = \frac{24}{25}}$. But hold on a second! We know that sine is the ratio of the opposite side (y) to the hypotenuse (r), or ${\frac{y}{r}}$. In our case, ${y = -24}$ and ${r = 25}$, so ${\sin(\theta) = \frac{-24}{25}}$. The student missed the negative sign! This is a critical error because it changes the entire nature of the solution. Sine is negative in the third quadrant, so this oversight is a major red flag. Now, let's look at cosecant, which is the reciprocal of sine. The student has ${\csc(\theta) = \frac{25}{24}}$, but again, they've missed the negative sign. The correct value should be ${\csc(\theta) = \frac{25}{-24} = -\frac{25}{24}}$. This error highlights the importance of remembering the signs of trigonometric functions in different quadrants. Failing to account for the quadrant can lead to incorrect answers, even if the numerical values are close. This slip-up underscores the interconnectedness of sine and cosecant. If one is wrong, the other is bound to be wrong as well. So, let's keep those quadrant rules in mind and ensure our signs are always on point! This careful attention to detail is what separates a good trigonometric solution from a flawed one.

Cosine and Secant A Tale of Reciprocal Negatives

Moving on, the student correctly stated that ${\cos(\theta) = -\frac{7}{25}}$. Awesome! Cosine is the ratio of the adjacent side (x) to the hypotenuse (r), which is ${\frac{x}{r}}$. With ${x = -7}$ and ${r = 25}$, we get ${\cos(\theta) = \frac{-7}{25}}$, which matches the student's answer. Now, let's check secant, which is the reciprocal of cosine. The student has ${\sec(\theta) = -\frac{25}{7}}$, and that's also correct! Flipping the fraction of cosine gives us secant, and the negative sign stays put because we are still in the third quadrant where cosine and secant are negative. This part of the solution demonstrates a good understanding of reciprocal trigonometric functions and how signs behave. The fact that both cosine and secant are correct gives us confidence in the student's grasp of these particular concepts. It's a testament to the student's accuracy in applying the definitions of these trigonometric functions. However, we can't let our guard down yet! There are still other trigonometric values to check, and we need to maintain our meticulous approach to ensure we catch any other potential errors.

Tangent and Cotangent The Positive Duo

Now, let’s dive into tangent and cotangent. The student has ${\tan(\theta) = \frac{24}{7}}$. Tangent is the ratio of the opposite side (y) to the adjacent side (x), or ${\frac{y}{x}}$. In our case, ${y = -24}$ and ${x = -7}$, so ${\tan(\theta) = \frac{-24}{-7} = \frac{24}{7}}$. This is correct! The negative signs cancel out, giving us a positive value, which is expected in the third quadrant. Moving on to cotangent, which is the reciprocal of tangent, the student has ${\cot(\theta) = \frac{7}{24}}$. This is also correct! We simply flip the fraction of tangent, and since tangent is positive, cotangent is also positive. This shows a solid understanding of how tangent and cotangent relate to each other and how they behave in the third quadrant. The student has correctly applied the definitions and reciprocal relationships here. This is encouraging because it highlights a strong area of competence in their trigonometric calculations. However, we must remember that consistency across all trigonometric functions is key, so we still need to be vigilant for any other potential errors. Let's keep this positive momentum going as we wrap up our analysis!

Correcting the Course Setting the Trigonometric Values Straight

Alright, guys, we’ve identified the errors! The student mixed up the signs for sine and cosecant. Let’s set the record straight and present the correct trigonometric values:

• ${\sin(\theta) = -\frac{24}{25}}$
• ${\csc(\theta) = -\frac{25}{24}}$
• ${\cos(\theta) = -\frac{7}{25}}$
• ${\sec(\theta) = -\frac{25}{7}}$
• ${\tan(\theta) = \frac{24}{7}}$
• ${\cot(\theta) = \frac{7}{24}}$

These values accurately reflect the relationships between the sides of the triangle and the angle ${\theta}$ in the third quadrant. By carefully correcting the signs for sine and cosecant, we’ve ensured that our solution aligns with trigonometric principles and quadrant rules. This revised set of values presents a complete and accurate picture of the trigonometric functions for the given angle. It's a testament to the importance of meticulousness and a deep understanding of trigonometric concepts. This final set of answers provides clarity and correctness, solidifying our understanding of this problem.

Key Takeaways Mastering Trigonometric Functions

So, what have we learned today? Firstly, always, always, always consider the quadrant! The signs of trigonometric functions are heavily dependent on the quadrant in which the angle lies. Secondly, remember the definitions of the trigonometric functions and their reciprocals. Sine and cosecant are reciprocals, cosine and secant are reciprocals, and tangent and cotangent are reciprocals. Lastly, practice makes perfect! The more you work with these concepts, the more natural they will become. Guys, trigonometry might seem tricky at first, but with a clear understanding of the basics and a systematic approach, you can conquer any trigonometric puzzle. Keep up the great work, and remember, math is an adventure! This exercise underscores the significance of attention to detail and the power of a systematic approach in solving trigonometric problems. By breaking down the problem into smaller, manageable steps, we can identify and correct errors effectively. This experience reinforces the idea that understanding the underlying principles of trigonometry is crucial for accurate problem-solving. So, let’s embrace the challenges that trigonometry presents and continue to hone our skills, knowing that each problem solved brings us closer to mastery!

This problem shows the importance of careful analysis in trigonometry. A simple sign error can throw off the entire solution. By understanding the relationships between trigonometric functions and the quadrants, we can avoid these mistakes and solve problems with confidence.