Find Tan(A) In Quadrant IV: A Trig Problem Solved

by Omar Yusuf 50 views

Hey guys! Let's dive into a trigonometry problem where we need to find the value of tan(A) given cos(A) and the quadrant it lies in. We'll use some fundamental trigonometric identities to solve this. So, buckle up, and let's get started!

Understanding the Problem

Okay, so here's the deal. We know that cos(A) = 0.325, and we need to find tan(A). But there's a catch! We also know that angle A is in the fourth quadrant. This is super important because it tells us about the signs of our trigonometric functions. Remember the CAST rule? It’s a lifesaver here! CAST (Cosine, All, Sine, Tangent) tells us which trig functions are positive in each quadrant. In the fourth quadrant, only cosine is positive. That means sine and tangent are negative. This is a crucial detail that will help us get the correct answer.

We are also given two trigonometric identities:

  1. The Pythagorean identity: sin²(A) + cos²(A) = 1. This is like the backbone of trigonometry – we use it all the time!
  2. The quotient identity: tan(A) = sin(A) / cos(A). This one directly links sine, cosine, and tangent, which is exactly what we need.

Our plan is simple: First, we'll use the Pythagorean identity to find sin(A). Then, we'll use the quotient identity to calculate tan(A). Finally, we'll make sure our answer has the correct sign based on the quadrant information.

Step 1: Finding Sin(A) Using the Pythagorean Identity

Alright, let’s roll up our sleeves and get into the math. We know that sin²(A) + cos²(A) = 1. We also know that cos(A) = 0.325. So, we can plug that value into our identity:

sin²(A) + (0.325)² = 1

First, we calculate (0.325)²:

(0.325)² = 0.105625

Now our equation looks like this:

sin²(A) + 0.105625 = 1

Next, we want to isolate sin²(A), so we subtract 0.105625 from both sides of the equation:

sin²(A) = 1 - 0.105625

sin²(A) = 0.894375

Now, to find sin(A), we need to take the square root of both sides:

sin(A) = ±√0.894375

Using a calculator, we find:

√0.894375 ≈ 0.945714

So, sin(A) = ±0.945714. But hold on! We're not done yet. Remember, we know that angle A is in the fourth quadrant, and in the fourth quadrant, sine is negative. Therefore, we choose the negative value:

sin(A) = -0.945714

Step 2: Finding Tan(A) Using the Quotient Identity

Great! We've found sin(A). Now we can use the quotient identity, tan(A) = sin(A) / cos(A), to find tan(A). We have sin(A) = -0.945714 and cos(A) = 0.325, so we plug these values into the equation:

tan(A) = -0.945714 / 0.325

Now, we just divide:

tan(A) ≈ -2.910043

Step 3: Rounding to Ten-Thousandths

The problem asks us to round our answer to the ten-thousandths place. That's four decimal places. So, we look at the fifth decimal place to decide whether to round up or down. In our case, we have:

tan(A) ≈ -2.910043

The fifth decimal place is a 4, which is less than 5, so we round down and keep the fourth decimal place as it is:

tan(A) ≈ -2.9100

Conclusion

And there we have it! We've successfully found tan(A) using the given information and trigonometric identities. By applying the Pythagorean identity, the quotient identity, and considering the quadrant in which the angle lies, we determined that tan(A) ≈ -2.9100. Remember, guys, always pay attention to the signs of trigonometric functions in different quadrants – it can make or break your answer! Understanding the CAST rule and the fundamental trigonometric identities are key to solving these problems. Keep practicing, and you'll become a trig whiz in no time!

So, the final answer, rounded to ten-thousandths, is approximately -2.9100. It's always a good idea to double-check your work, especially with these kinds of problems, to make sure you haven't made any simple errors. But overall, the process is straightforward once you understand the underlying principles.