Trigonometric Functions In Quadrant IV: Finding All Values

Hey guys! Let's dive into a fun trigonometric problem today. We're given that angle θ is chilling in Quadrant IV, and we know that sec θ = -7/4. Our mission, should we choose to accept it, is to find the values of all the other trigonometric functions. Don't worry; it's not as daunting as it sounds! We will explore how to find x, y, r, sin θ, cos θ, and tan θ in this scenario. So grab your calculators and let's get started!

Understanding the Problem

Before we jump into calculations, let's break down what we know and what we need to find. The problem tells us that angle θ is in Quadrant IV. This is super important because it tells us about the signs of our x and y coordinates. Remember, in Quadrant IV, x is positive, and y is negative. This will be crucial when we determine the signs of our trigonometric functions. We're also given that sec θ = -7/4. Secant is the reciprocal of cosine, so this means cos θ = -4/7. However, since cosine is defined as x/r, and r (the radius) is always positive, the negative sign must apply to the x-coordinate. Keep this in mind! The main goal here is to find the values of all six trigonometric functions: sine (sin θ), cosine (cos θ), tangent (tan θ), cosecant (csc θ), secant (sec θ), and cotangent (cot θ). We already have sec θ, so we're part of the way there. To find the rest, we need to figure out the values of x, y, and r, which represent the coordinates and the radius of the point where the terminal side of angle θ intersects the unit circle (or a circle of radius r).

Thinking about it visually can really help. Imagine a circle centered at the origin. Quadrant IV is the bottom-right quadrant. Our angle θ starts at the positive x-axis and rotates clockwise into Quadrant IV. The point where the terminal side of this angle hits the circle has a positive x-coordinate and a negative y-coordinate. This is our key to solving the problem. Remembering these fundamental relationships and the characteristics of each quadrant is essential for tackling trigonometric problems with confidence. We will use these concepts to systematically find the missing values and complete our trigonometric picture. Let's move on to finding those values!

Finding x, y, and r

Okay, let’s get our hands dirty with some calculations! We know that sec θ = -7/4, and we've established that this means cos θ = -4/7. Since cosine is defined as x/r, we can say that x = -4 and r = 7. Wait a minute! Remember how we said x should be positive in Quadrant IV? This is a bit of a trick question! The secant function is negative in Quadrants II and III, but since our angle is in Quadrant IV, we need to consider the relationship between secant and cosine carefully. While secant is the reciprocal of cosine, the sign of cosine in Quadrant IV is actually positive. Therefore, there seems to be a mistake in the problem statement. Secant cannot be negative in the fourth quadrant. Assuming the intention was for sec θ to be a positive value, we'll proceed with sec θ = 7/4, making cos θ = 4/7. Thus, x = 4 and r = 7.

Now that we have x and r, we can find y using the Pythagorean theorem: x² + y² = r². Plugging in our values, we get 4² + y² = 7², which simplifies to 16 + y² = 49. Subtracting 16 from both sides gives us y² = 33. Taking the square root of both sides, we get y = ±√33. Remember that in Quadrant IV, y is negative, so we choose the negative root: y = -√33. Awesome! We've found our x, y, and r values: x = 4, y = -√33, and r = 7. With these values, we can now calculate the remaining trigonometric functions. This is where all our hard work pays off, and we can see how the coordinates and the radius connect to the trigonometric ratios. So let's move on to the exciting part of calculating sine, cosine, tangent, and their reciprocals!

Calculating the Other Trigonometric Functions

Alright, guys, now for the grand finale! We've got our x, y, and r values (x = 4, y = -√33, and r = 7), and we're ready to find the other trigonometric functions. Let's start with sine (sin θ). Sine is defined as y/r, so sin θ = (-√33)/7. Next up is cosine (cos θ), which we already found when we corrected the problem (cos θ = 4/7). Now let's tackle tangent (tan θ). Tangent is defined as y/x, so tan θ = (-√33)/4. We've got the main three! Now for their reciprocals.

Cosecant (csc θ) is the reciprocal of sine, so csc θ = 1/sin θ = -7/√33. To rationalize the denominator, we multiply both the numerator and denominator by √33, giving us csc θ = (-7√33)/33. Secant (sec θ) is the reciprocal of cosine, which we already know is 7/4 (from our corrected assumption). Finally, cotangent (cot θ) is the reciprocal of tangent, so cot θ = 1/tan θ = -4/√33. Again, we rationalize the denominator by multiplying both the numerator and denominator by √33, resulting in cot θ = (-4√33)/33. And there we have it! We've found all six trigonometric functions: sin θ = (-√33)/7, cos θ = 4/7, tan θ = (-√33)/4, csc θ = (-7√33)/33, sec θ = 7/4, and cot θ = (-4√33)/33. We took a potentially confusing problem (negative secant in Quadrant IV) and turned it into a triumph by carefully considering the relationships between trigonometric functions and the geometry of the coordinate plane.

Final Values

To recap, here are the values we found:

• x = 4
• y = -√33
• r = 7
• sin θ = (-√33)/7
• cos θ = 4/7
• tan θ = (-√33)/4

Conclusion

So, guys, we've successfully navigated this trigonometric challenge! We started by understanding the problem, identifying the quadrant, and using the given information to find x, y, and r. Then, we used those values to calculate all six trigonometric functions. Remember, the key to these problems is understanding the relationships between the functions and the geometry of the coordinate plane. Keep practicing, and you'll become a trig wizard in no time! This exercise highlights the importance of attention to detail and a solid grasp of fundamental trigonometric principles. Remember to always double-check the signs of trigonometric functions in different quadrants and to carefully consider the relationships between the functions and their reciprocals. Great job, and keep exploring the fascinating world of trigonometry!