Finding K And Α For Point P(-4) In Third Quadrant
In the fascinating realm of coordinate geometry and trigonometry, a seemingly simple problem can often unlock a wealth of mathematical understanding. Today, we embark on a journey to explore such a problem, one that involves a point P(k, -4) nestled in the third quadrant, an angle α (∠NOP), and a line segment OP with a length of 17 units. Our mission is to unravel the mysteries surrounding this geometric configuration and calculate the value of k and the size of α, all while adhering to the precision of two decimal places.
3.2.1 Finding the Value of k: A Symphony of the Pythagorean Theorem and Quadrant Awareness
To determine the value of k, we must first recognize the geometric orchestra at play. We have a right-angled triangle formed by the x-axis, the y-axis, and the line segment OP. The coordinates of point P(k, -4) provide us with the lengths of the two legs of this triangle: |k| (the absolute value of k) along the x-axis and 4 units along the y-axis. The length of the hypotenuse, OP, is given as 17 units. This is where the Pythagorean Theorem, a cornerstone of geometry, enters our stage. The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In our case, this translates to:
OP² = k² + (-4)²
Substituting the given value of OP, we get:
17² = k² + 16
289 = k² + 16
Now, we isolate k²:
k² = 289 - 16
k² = 273
To find k, we take the square root of both sides:
k = ±√273
Here, the quadrant awareness becomes crucial. We are told that point P(k, -4) lies in the third quadrant. In the third quadrant, both the x and y coordinates are negative. Therefore, we discard the positive solution and embrace the negative one:
k = -√273
Calculating the square root of 273 and rounding to two decimal places, we arrive at:
k ≈ -16.52
Thus, the value of k is approximately -16.52. This seemingly simple calculation has unveiled a crucial piece of our geometric puzzle, showcasing the power of the Pythagorean Theorem and the importance of quadrant awareness in coordinate geometry. Understanding the relationship between the coordinates of a point and its position in a quadrant is paramount in solving problems of this nature. The negative value of k reaffirms our understanding of the third quadrant, where x-coordinates reside in the negative realm. This meticulous approach, blending geometric principles with careful consideration of signs, allows us to navigate the intricacies of coordinate geometry with confidence and precision.
3.2.3 Determining the Size of α (∠NOP): A Trigonometric Tango
Now, let's turn our attention to the angle α (∠NOP). To find the size of α, we venture into the realm of trigonometry. In our right-angled triangle, we have the lengths of all three sides: the opposite side (4 units), the adjacent side (|k| ≈ 16.52 units), and the hypotenuse (17 units). We can employ any of the trigonometric ratios – sine, cosine, or tangent – to relate α to these side lengths. However, to minimize the risk of error propagation (in case our calculated value of k has a slight deviation), it's often wise to use the given values whenever possible. In this case, we know the opposite side (4) and the hypotenuse (17), making the sine function a suitable candidate.
sin(α') = Opposite / Hypotenuse = 4 / 17
Here, α' represents the reference angle, the acute angle formed between the terminal arm of α and the x-axis. To find α', we take the inverse sine (arcsin) of 4/17:
α' = arcsin(4/17)
Using a calculator, we find:
α' ≈ 13.66°
However, this is not the final answer. We must remember that α lies in the third quadrant. The reference angle α' is the angle formed with the x-axis, but α is measured from the positive x-axis in a counter-clockwise direction. In the third quadrant, the angle α is given by:
α = 180° + α'
Substituting the value of α', we get:
α ≈ 180° + 13.66°
α ≈ 193.66°
Therefore, the size of α is approximately 193.66 degrees. This journey into the trigonometric realm showcases the interplay between angles and side lengths in a right-angled triangle. By carefully selecting the appropriate trigonometric ratio and considering the quadrant in which the angle lies, we can accurately determine its size. The concept of a reference angle is crucial in this process, allowing us to bridge the gap between the acute angle obtained from the inverse trigonometric function and the actual angle in the specified quadrant. This meticulous approach, combining trigonometric principles with a keen awareness of quadrant properties, empowers us to navigate the intricacies of angle calculations with confidence and precision.
Conclusion: A Symphony of Geometry and Trigonometry
In conclusion, by skillfully applying the Pythagorean Theorem and trigonometric principles, we have successfully calculated the value of k and the size of α in the given geometric configuration. We found that k is approximately -16.52 and α is approximately 193.66 degrees. This exercise not only demonstrates the power of these mathematical tools but also highlights the importance of careful consideration of quadrant properties and the relationships between angles and side lengths in a triangle. The interplay between geometry and trigonometry is a beautiful testament to the interconnectedness of mathematical concepts. By mastering these fundamental principles, we unlock a world of problem-solving possibilities, empowering us to tackle even more complex geometric and trigonometric challenges with confidence and precision.
This exploration serves as a reminder that mathematics is not merely a collection of formulas and equations, but a powerful language for describing and understanding the world around us. Each problem we solve, each theorem we prove, contributes to a deeper appreciation of the elegance and interconnectedness of mathematical ideas. As we continue our mathematical journey, let us embrace the challenges, celebrate the discoveries, and marvel at the beauty of mathematics.