Finding Trigonometric Functions For Point (-4, 7) A Step-by-Step Guide

Introduction

In trigonometry, one of the fundamental concepts is finding the values of trigonometric functions for a given angle. When we have a point on the terminal side of an angle in standard position, we can determine the exact values of all six trigonometric functions. This article will guide you through the process of finding these values, specifically when given the point (-4, 7). Understanding these functions is crucial for various applications in mathematics, physics, and engineering. The six trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Each of these functions relates the sides of a right triangle to its angles. By understanding these relationships, we can solve a wide range of problems involving angles and distances. To effectively grasp this concept, it's essential to start with the basics: the definitions of the trigonometric functions and how they relate to the Cartesian coordinate system. Let's delve into the steps required to calculate these trigonometric values accurately, providing a comprehensive guide for students and enthusiasts alike. The process involves using the coordinates of the given point to find the distance from the origin, also known as the radius, and then applying the definitions of the trigonometric functions. This approach ensures a clear and systematic method for solving such problems.

Understanding Trigonometric Functions

Before diving into the specifics, it’s important to understand the basic definitions of the six trigonometric functions. Consider a point (x, y) on the terminal side of an angle θ in standard position. The distance from the origin to this point, denoted as r, can be found using the Pythagorean theorem: r = √(x² + y²). The trigonometric functions are defined as follows:

• Sine (sin θ): The ratio of the opposite side (y) to the hypotenuse (r). sin θ = y/r
• Cosine (cos θ): The ratio of the adjacent side (x) to the hypotenuse (r). cos θ = x/r
• Tangent (tan θ): The ratio of the opposite side (y) to the adjacent side (x). tan θ = y/x
• Cosecant (csc θ): The reciprocal of sine. csc θ = r/y
• Secant (sec θ): The reciprocal of cosine. sec θ = r/x
• Cotangent (cot θ): The reciprocal of tangent. cot θ = x/y

These definitions form the cornerstone of trigonometric calculations. By understanding these ratios, we can determine the values of the trigonometric functions for any angle, provided we have a point on its terminal side. The signs of these functions in different quadrants are determined by the signs of x and y. For instance, in the first quadrant (where both x and y are positive), all six trigonometric functions are positive. In the second quadrant (where x is negative and y is positive), sine and cosecant are positive, while the others are negative. This understanding is critical for accurately determining the values of trigonometric functions in various scenarios. Knowing the definitions and how they apply in different quadrants will enable us to solve more complex trigonometric problems effectively. The reciprocal relationships between the functions also simplify calculations, as knowing one function's value can immediately give us the value of its reciprocal.

Step-by-Step Solution for the Point (-4, 7)

Given the point (-4, 7) on the terminal side of θ, we will now find the exact values of the six trigonometric functions. This process involves several steps, each building upon the previous one to ensure accuracy and clarity. We'll start by calculating the distance from the origin to the point, then use this distance to find the values of the trigonometric functions.

Step 1: Calculate r

First, we need to find the distance r from the origin to the point (-4, 7). Using the formula r = √(x² + y²), we substitute x = -4 and y = 7:

r = √((-4)² + (7)²) = √(16 + 49) = √65

Thus, the distance r is √65. This value will be used as the denominator in our sine and cosine calculations and as the numerator in our cosecant and secant calculations. Understanding the significance of r is crucial, as it represents the hypotenuse of the right triangle formed by the point, the origin, and the x-axis. This hypotenuse is the foundation for calculating all six trigonometric ratios. Correctly calculating r is essential because an error here will propagate through all subsequent calculations, leading to incorrect values for the trigonometric functions. The square root of 65 cannot be simplified further, so we will leave it in this form for exact calculations. This step highlights the importance of precision in mathematical calculations and sets the stage for the rest of the problem.

Step 2: Determine the Trigonometric Function Values

Now that we have the value of r, we can determine the six trigonometric function values using their definitions:

• Sine (sin θ):

  sin θ = y/r = 7/√65. Rationalizing the denominator, we multiply the numerator and denominator by √65:

  sin θ = (7√65) / 65

  The sine function relates the y-coordinate to the distance from the origin. In this case, it represents the ratio of the opposite side to the hypotenuse in the right triangle formed. Rationalizing the denominator is a standard practice to simplify the expression and make it easier to work with in further calculations. The exact value of sine, with a rationalized denominator, is crucial for accurate problem-solving.

• Cosine (cos θ):

  cos θ = x/r = -4/√65. Rationalizing the denominator:

  cos θ = (-4√65) / 65

  The cosine function relates the x-coordinate to the distance from the origin. It is the ratio of the adjacent side to the hypotenuse. The negative sign is important here, indicating that the angle θ is in a quadrant where cosine is negative. Understanding the sign of the trigonometric functions in different quadrants is essential for accurate calculations.

• Tangent (tan θ):

  tan θ = y/x = 7 / -4 = -7/4

  The tangent function is the ratio of the sine to the cosine or, directly, the ratio of the y-coordinate to the x-coordinate. In this case, it represents the slope of the line from the origin to the point (-4, 7). The negative sign indicates that the line has a negative slope, which aligns with the point being in the second quadrant.

• Cosecant (csc θ):

  csc θ = r/y = √65 / 7

  Cosecant is the reciprocal of sine. It’s straightforward to find once we have the value of sine. This function is particularly useful in applications where we need to relate the hypotenuse to the opposite side.

• Secant (sec θ):

  sec θ = r/x = √65 / -4 = (-√65) / 4

  Secant is the reciprocal of cosine. Again, the negative sign is crucial, reflecting that secant is negative in the second quadrant.

• Cotangent (cot θ):

  cot θ = x/y = -4/7

  Cotangent is the reciprocal of tangent. It represents the ratio of the adjacent side to the opposite side. The negative sign is consistent with the tangent value and the quadrant in which the angle lies.

Step 3: Summarize the Results

To summarize, the exact values of the six trigonometric functions for the point (-4, 7) are:

• sin θ = (7√65) / 65
• cos θ = (-4√65) / 65
• tan θ = -7/4
• csc θ = √65 / 7
• sec θ = (-√65) / 4
• cot θ = -4/7

These values provide a comprehensive understanding of the trigonometric ratios for the angle θ formed by the point (-4, 7) in standard position. Each function's value gives a unique perspective on the angle's properties, which can be used in various mathematical and real-world applications. The consistency of the signs of the functions across different quadrants is a crucial check for the accuracy of the calculations. For example, in the second quadrant, sine and cosecant are positive, while the others are negative, which aligns with our calculated values.

Common Mistakes to Avoid

When calculating trigonometric functions, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them and ensure accuracy. One frequent error is incorrectly applying the Pythagorean theorem when finding r. Double-checking the signs and squaring each term correctly is crucial. Forgetting to rationalize the denominator is another common mistake, especially when dealing with square roots. Always remember to multiply both the numerator and denominator by the square root to eliminate it from the denominator.

Another significant error involves mixing up the definitions of the trigonometric functions. A mnemonic, such as SOH-CAH-TOA (Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent), can be helpful. Similarly, remembering that cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent, respectively, is essential. Pay close attention to the signs of the trigonometric functions in different quadrants. For instance, sine is positive in the first and second quadrants, while cosine is positive in the first and fourth quadrants. Tangent is positive in the first and third quadrants. A sign error can completely change the result and the interpretation of the problem.

Finally, a simple arithmetic mistake during the calculation can lead to a wrong answer. It is always a good idea to double-check each step, especially when dealing with fractions and square roots. Practice and familiarity with these calculations can significantly reduce the likelihood of errors. By being mindful of these common mistakes and taking the necessary precautions, you can improve your accuracy and confidence in solving trigonometric problems.

Conclusion

In conclusion, finding the exact values of the six trigonometric functions given a point on the terminal side of an angle involves a systematic process. We start by calculating the distance r from the origin to the point using the Pythagorean theorem. Then, we apply the definitions of sine, cosine, tangent, cosecant, secant, and cotangent to find their values. Rationalizing the denominator, when necessary, simplifies the expressions and provides a standard form for the answers. Throughout this process, understanding the definitions of the functions and the signs in different quadrants is essential for accuracy. By following these steps carefully and avoiding common mistakes, you can confidently determine the exact trigonometric values for any given point. Mastering these concepts is fundamental for further studies in trigonometry and its applications in various fields. The ability to accurately calculate trigonometric functions is not only crucial for academic success but also for practical problem-solving in real-world scenarios. The consistent application of these principles will enhance your mathematical skills and deepen your understanding of trigonometry.