Finding Tan A In Quadrant IV When Sin A Is -0.78

Hey guys! Ever find yourself wrestling with trig functions and identities? It can feel like trying to solve a puzzle with missing pieces, right? Let's break down one of those puzzles today, focusing on how to find the tangent of an angle when we know its sine and which quadrant it lives in. We're going to use some fundamental trigonometric identities and a bit of logical deduction to crack this one. Stick with me, and you'll see it's not as daunting as it looks! This article aims to provide a comprehensive guide on how to solve for ${\tan(A)}$ when ${\sin(A) = -0.78}$ in the fourth quadrant. We'll walk through each step, ensuring you understand the underlying principles and can apply them to similar problems.

The Trigonometric Challenge

Let's dive right into the heart of our problem. We're given that ${\sin(A) = -0.78}$, and we need to find ${\tan(A)}$. The twist? We know that angle A is chilling out in the fourth quadrant. This quadrant clue is super important because it tells us about the signs of our trigonometric functions. Remember, in the fourth quadrant, cosine is positive, while sine and tangent are negative. Knowing this will help us make sure our final answer makes sense. In tackling this trigonometric problem, we are presented with a classic scenario that tests our understanding of trigonometric identities and quadrant rules. Our goal is to determine the value of ${\tan(A)}$ given that ${\sin(A) = -0.78}$ and that A lies in the fourth quadrant. This requires us to use the Pythagorean identity, ${\sin^2(A) + \cos^2(A) = 1}$, to find ${\cos(A)}$, and then apply the quotient identity, ${\tan(A) = \frac{\sin(A)}{\cos(A)}}$. The problem not only assesses our ability to manipulate these identities but also our understanding of how the signs of trigonometric functions vary across different quadrants. Let's embark on this mathematical journey together, breaking down each step to ensure clarity and comprehension. We'll start by revisiting the fundamental trigonometric identities that will serve as the cornerstone of our solution, ensuring we're all on the same page before we delve into the calculations.

Essential Trigonometric Identities

Before we jump into calculations, let's quickly recap the trig identities we'll be using. These are the bread and butter of solving trig problems, so having them handy is key. The first identity we'll use is the Pythagorean identity: $\sin^2(A) + \cos^2(A) = 1$ This is like the superhero of trig identities – it pops up everywhere! It relates sine and cosine, which is exactly what we need since we know sine and want to find cosine. The second identity is the quotient identity for tangent: $ an(A) = \frac\sin(A)}{\cos(A)}$ This one is straightforward but super powerful. Once we have sine and cosine, we can easily find tangent. Understanding these identities is like having the right tools in your toolbox – you can't build anything without them! These trigonometric identities are not just abstract formulas; they are the foundational principles that govern the relationships between different trigonometric functions. The Pythagorean identity, in particular, is a cornerstone of trigonometry, linking the squares of sine and cosine to unity. It stems directly from the Pythagorean theorem applied to the unit circle, illustrating the fundamental connection between geometry and trigonometry. Similarly, the quotient identity for tangent provides a direct link between sine, cosine, and tangent, allowing us to calculate tangent once we know sine and cosine. Grasping the origins and implications of these identities is crucial for developing a deep understanding of trigonometry and for effectively solving problems like the one at hand. Now that we've refreshed our memory on these essential tools, let's move on to the next step using the Pythagorean identity to find the value of ${\cos(A)$.

Step-by-Step Solution

1. Finding Cosine Using the Pythagorean Identity

Okay, let's put our first identity to work. We know ${\sin(A) = -0.78}$, so let's plug that into the Pythagorean identity:$\sin^2(A) + \cos^2(A) = 1$ $(-0.78)^2 + \cos^2(A) = 1$ Now, let's do the math:$0.6084 + \cos^2(A) = 1$ Subtract 0.6084 from both sides:$\cos^2(A) = 1 - 0.6084$ $\cos^2(A) = 0.3916$ Now, we take the square root of both sides to solve for ${\cos(A)}$:$\cos(A) = \pm \sqrt0.3916}$ $\cos(A) = \pm 0.6258$ Here's where our quadrant knowledge comes in! Since we're in the fourth quadrant, cosine is positive. So, we choose the positive root$\cos(A) = 0.6258$ We've nailed the first part – we found ${\cos(A)$! Applying the Pythagorean identity is a crucial step in solving this problem. By substituting the given value of ${\sin(A)}$ into the identity, we've successfully isolated ${\cos^2(A)}$. The subsequent steps involve basic algebraic manipulations: squaring the sine value, subtracting it from 1, and then taking the square root to find ${\cos(A)}$. However, the crucial part is the consideration of the quadrant in which angle A lies. Since cosine is positive in the fourth quadrant, we correctly choose the positive square root. This highlights the importance of understanding quadrant rules in trigonometry. Without this knowledge, we might end up with the wrong sign for ${\cos(A)}$, leading to an incorrect final answer. Now that we've accurately determined the value of ${\cos(A)}$, we're one step closer to finding ${\tan(A)}$. Next, we'll use the quotient identity to bring it all home.

2. Calculating Tangent Using the Quotient Identity

Alright, we're on the home stretch! Now that we know both ${\sin(A)}$ and ${\cos(A)}$, we can use the quotient identity to find ${\tan(A)}$. Remember, the quotient identity is:$\tan(A) = \frac\sin(A)}{\cos(A)}$ We have ${\sin(A) = -0.78}$ and ${\cos(A) = 0.6258}$, so let's plug those values in$\tan(A) = \frac{-0.780.6258}$ Now, we just divide$\tan(A) = -1.2464$ We've got our answer! The tangent of angle A is approximately -1.2464. Notice that the tangent is negative, which makes sense since tangent is negative in the fourth quadrant. This confirms that our answer is in the right ballpark. This step demonstrates the direct application of the quotient identity, a fundamental relationship between tangent, sine, and cosine. By substituting the values we've previously calculated for ${\sin(A)$ and ${\cos(A)}$, we arrive at the value of ${\tan(A)}$. The negative sign of the result is consistent with the fact that tangent is negative in the fourth quadrant, which serves as a useful check on our work. This reinforces the importance of understanding quadrant rules and how they affect the signs of trigonometric functions. The calculation itself is straightforward division, but the significance lies in the proper application of the identity and the correct interpretation of the sign. With this final calculation, we've successfully solved the problem. Let's recap our steps and highlight the key takeaways from this exercise.

3. Rounding to Ten-Thousandth

The problem asked us to round to the nearest ten-thousandth, which we've already done! Our answer, -1.2464, is rounded to four decimal places, so we're good to go. Rounding to the ten-thousandth place ensures we provide a precise answer, as requested in the problem statement. This step is a simple yet crucial part of the problem-solving process, emphasizing the importance of paying attention to detail and adhering to the given instructions. In practical applications, such precision might be necessary for accurate calculations and modeling. The ten-thousandth place represents a high degree of accuracy, illustrating the level of detail that trigonometric calculations can sometimes require. Now that we've rounded our answer, let's take a moment to summarize the entire solution process and highlight the key concepts we've utilized.

Conclusion: Tying It All Together

So, there you have it! We successfully found ${\tan(A)}$ using trig identities and our knowledge of quadrants. We started with ${\sin(A) = -0.78}$, used the Pythagorean identity to find ${\cos(A) = 0.6258}$, and then applied the quotient identity to get ${\tan(A) = -1.2464}$. The key takeaways here are:Trigonometric identities are your best friends. Memorize them and know how to use them. Quadrant information is crucial. It tells you the signs of your trig functions. Don't forget to round appropriately! Trigonometry might seem tricky at first, but with practice, you'll become a pro in no time! We've demonstrated how to tackle a typical trigonometric problem by systematically applying trigonometric identities and quadrant rules. The solution involves a series of logical steps, each building upon the previous one. We started by using the Pythagorean identity to find ${\cos(A)}$, emphasizing the importance of considering the quadrant to determine the correct sign. Then, we applied the quotient identity to calculate ${\tan(A)}$, again verifying that the sign of our answer aligns with the quadrant in which angle A lies. Throughout the process, we've highlighted the interconnectedness of trigonometric functions and the power of trigonometric identities in solving problems. This example serves as a testament to the elegance and efficiency of trigonometric methods in dealing with angular relationships. As we conclude, let's encourage further exploration and practice to solidify these concepts and build confidence in tackling more complex trigonometric challenges. Understanding these fundamental concepts and practicing similar problems will build your confidence and skills in trigonometry. Remember, math is like a muscle – the more you use it, the stronger it gets! Keep practicing, and you'll be solving trig problems like a boss in no time.

Final Answer

The final answer, rounded to ten-thousandth, is:$\tan(A) = -1.2464$

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