Finding Trigonometric Values Cosec Θ, Sec Θ, And Cot Θ For Point (-7,5)
In trigonometry, understanding the relationship between points on the coordinate plane and trigonometric functions is crucial. When we are given a point on the terminal side of an angle θ, we can determine the exact values of trigonometric functions such as cosecant (cosec θ), secant (sec θ), and cotangent (cot θ). In this article, we will walk through the process of finding these values when given the point (-7, 5) on the terminal side of θ. This involves using the definitions of trigonometric functions in terms of x, y, and r (the distance from the origin to the point), and simplifying the results to their exact forms. By understanding these steps, you can apply this knowledge to other points and angles, deepening your understanding of trigonometry.
Understanding Trigonometric Functions and the Unit Circle
Before diving into the specifics of our problem, let's briefly review the definitions of trigonometric functions in the context of the coordinate plane. Consider a point (x, y) on the terminal side of an angle θ in standard position (i.e., the initial side is on the positive x-axis). The distance r from the origin to the point (x, y) is given by the Pythagorean theorem: r = √(x² + y²). The six trigonometric functions are then defined as follows:
- Sine (sin θ) = y/r
- Cosine (cos θ) = x/r
- Tangent (tan θ) = y/x
- Cosecant (cosec θ) = r/y
- Secant (sec θ) = r/x
- Cotangent (cot θ) = x/y
These definitions are fundamental to solving trigonometric problems. The unit circle, which is a circle with a radius of 1 centered at the origin, is particularly useful for visualizing these relationships. On the unit circle, r = 1, which simplifies the trigonometric functions to sin θ = y, cos θ = x, and tan θ = y/x. The reciprocal functions (cosec θ, sec θ, and cot θ) are then simply the reciprocals of sine, cosine, and tangent, respectively. Understanding these definitions and the role of the unit circle is essential for finding the exact values of trigonometric functions.
Step-by-Step Solution for Point (-7, 5)
Step 1: Identify the Coordinates
We are given the point (-7, 5) on the terminal side of the angle θ. Thus, we can identify x = -7 and y = 5. These coordinates are the foundation for finding the values of the trigonometric functions. The negative x-coordinate indicates that the point lies in either the second or third quadrant, while the positive y-coordinate places it in the first or second quadrant. Combining these, we know that the angle θ must terminate in the second quadrant. Knowing the quadrant helps us anticipate the signs of the trigonometric functions; in the second quadrant, sine and cosecant will be positive, while cosine, secant, tangent, and cotangent will be negative.
Step 2: Calculate the Distance r
To find the trigonometric function values, we first need to calculate the distance r from the origin to the point (-7, 5). This is done using the Pythagorean theorem: r = √(x² + y²). Substituting the given values, we get:
r = √((-7)² + (5)²) r = √(49 + 25) r = √74
Thus, the distance r is √74. This value is crucial as it will be used in the denominators (or numerators, for reciprocal functions) of our trigonometric ratios. The fact that r is the square root of a number means that we might need to rationalize denominators later, but for now, we keep it in this form and proceed with calculating the trigonometric functions.
Step 3: Calculate the Trigonometric Function Values
Now that we have the values of x, y, and r, we can calculate the exact values of the trigonometric functions cosec θ, sec θ, and cot θ. Using the definitions of these functions:
- Cosec θ = r/y = √74 / 5
- Sec θ = r/x = √74 / -7 = -√74 / 7
- Cot θ = x/y = -7 / 5
These are the exact values for the given trigonometric functions. Note the signs of the values: cosec θ is positive, while sec θ and cot θ are negative, which aligns with our earlier observation that the angle θ lies in the second quadrant. The expressions involve the square root of 74, which cannot be simplified further since 74 has no perfect square factors other than 1. The values are now in their simplest, exact forms, providing a precise representation of the trigonometric functions for the angle θ.
Step 4: Summarize the Results
To summarize, given the point (-7, 5) on the terminal side of angle θ, we have found the following exact values for the trigonometric functions:
- cosec θ = √74 / 5
- sec θ = -√74 / 7
- cot θ = -7 / 5
These values represent the trigonometric ratios for the angle θ that passes through the point (-7, 5). Each value is exact and simplified, providing a clear and precise answer to the problem. This process demonstrates how to use the coordinates of a point on the terminal side of an angle to determine trigonometric function values, a fundamental concept in trigonometry.
Alternative Methods and Considerations
While the direct application of trigonometric definitions is the most straightforward method, there are alternative approaches and considerations to keep in mind. One such method involves finding the reference angle. The reference angle is the acute angle formed by the terminal side of θ and the x-axis. By finding the reference angle, we can use trigonometric values for acute angles (which are often memorized or readily available) and then adjust the sign based on the quadrant in which θ lies. For example, in our case, we could find the reference angle α such that tan α = |5/-7| = 5/7, and then use trigonometric values for α. However, since we are asked for exact values and the reference angle might not correspond to a special angle (like 30°, 45°, or 60°), this method might not directly give us exact answers without additional calculations involving inverse trigonometric functions.
Another consideration is the domain and range of the trigonometric functions. Cosecant, secant, and cotangent have restrictions on their domains because they involve division. Specifically, cosec θ is undefined when y = 0, sec θ is undefined when x = 0, and cot θ is undefined when y = 0. These restrictions arise from the denominators in their respective definitions. In our problem, since y = 5 and x = -7, none of these functions are undefined, but it's crucial to be aware of these limitations when dealing with other points and angles.
Real-World Applications and Further Exploration
The concepts explored in this article have numerous real-world applications, particularly in fields such as physics, engineering, and navigation. Trigonometric functions are used to model periodic phenomena, such as the motion of waves and oscillations. They are also fundamental in vector analysis, which is essential for understanding forces and motion in two and three dimensions. In navigation, trigonometric functions are used to calculate distances and bearings, enabling accurate mapping and positioning.
To further explore this topic, consider investigating the graphs of trigonometric functions and how the unit circle relates to these graphs. Understanding the periodicity, amplitude, and phase shifts of sine, cosine, and tangent (and their reciprocals) will deepen your comprehension of trigonometric functions. Additionally, exploring trigonometric identities and their applications can provide more tools for solving complex trigonometric problems. By continuing to build your knowledge in these areas, you will strengthen your ability to apply trigonometry in various mathematical and real-world contexts.
Conclusion
In conclusion, we have successfully determined the exact values of cosec θ, sec θ, and cot θ for the angle θ whose terminal side passes through the point (-7, 5). By understanding the definitions of trigonometric functions and applying the Pythagorean theorem, we found that cosec θ = √74 / 5, sec θ = -√74 / 7, and cot θ = -7 / 5. This process demonstrates the fundamental connection between points on the coordinate plane and trigonometric values. The ability to find exact values of trigonometric functions is a crucial skill in trigonometry, with applications ranging from solving geometric problems to modeling real-world phenomena. Continued practice and exploration of these concepts will further solidify your understanding and proficiency in trigonometry.
