Finding The Trigonometric Identity True For All Real X

Jul 16, 2025 by ADMIN 55 views

For All Real Values of x, Which Equation Holds True?

#title: Unraveling Trigonometric Identities Finding the True Equation for All Real Values of x

In the realm of mathematics, particularly trigonometry, lies a fascinating exploration of functions and their relationships. When faced with equations involving trigonometric functions like sine and cosine, it's crucial to understand their fundamental properties and identities. In this article, we delve into the question of identifying the equation that holds true for all real values of x, examining various options and employing trigonometric principles to arrive at the correct answer. This exploration will not only solidify your understanding of trigonometric identities but also enhance your problem-solving skills in mathematics.

Understanding the Basics of Trigonometric Functions

Before diving into the specific equations, let's refresh our understanding of the sine and cosine functions. These functions are fundamental to trigonometry and are defined based on the unit circle. For any angle θ, the coordinates of the point where the terminal side of the angle intersects the unit circle are given by (cos θ, sin θ). This geometric definition leads to several key properties:

Range: Both sine and cosine functions have a range of [-1, 1]. This means that for any real value of x, -1 ≤ sin(x) ≤ 1 and -1 ≤ cos(x) ≤ 1. This property is crucial for eliminating options that yield values outside this range.
Periodicity: Sine and cosine are periodic functions with a period of 2π. This means that their values repeat every 2π radians. This periodicity is essential for understanding the behavior of these functions over the entire real number line.
Pythagorean Identity: One of the most fundamental trigonometric identities is the Pythagorean identity: sin²(x) + cos²(x) = 1. This identity is derived directly from the Pythagorean theorem applied to the unit circle and is a cornerstone for solving trigonometric equations.

Understanding these basic properties is essential to navigate the problem effectively and identify the correct equation that holds true for all real values of x. With a solid foundation in these concepts, we can now dissect the given equations and determine their validity.

Analyzing the Given Equations

Now, let's analyze each of the given equations to determine which one holds true for all real values of x. We will use our understanding of the range of sine and cosine functions, as well as the Pythagorean identity, to evaluate each option.

F. sin(7x) + cos(7x) = 7

This equation can be quickly eliminated by considering the range of sine and cosine functions. As we established earlier, both sin(x) and cos(x) have a range of [-1, 1]. Therefore, for any real value of x, -1 ≤ sin(7x) ≤ 1 and -1 ≤ cos(7x) ≤ 1. The maximum possible value of sin(7x) + cos(7x) occurs when both sin(7x) and cos(7x) are at their maximum values, which is 1. Thus, the maximum value of sin(7x) + cos(7x) is 1 + 1 = 2. Since 2 is significantly less than 7, this equation cannot be true for all real values of x. The range limitation makes this option invalid.

G. sin(7x) + cos(7x) = 1

This equation is more nuanced than the previous one. While it's true that sin(7x) + cos(7x) can equal 1 for some values of x, it's not true for all values. To see why, consider the trigonometric identity: sin(a + b) = sin(a)cos(b) + cos(a)sin(b). We can rewrite the left side of the equation using this identity, but it won't simplify to a constant value for all x. We can test specific values of x to see if the equation holds. For instance, if x = 0, then sin(0) + cos(0) = 0 + 1 = 1, which satisfies the equation. However, if we choose a different value, such as x = π/14, then sin(π/2) + cos(π/2) = 1 + 0 = 1, which also works. But this doesn't guarantee it's true for all x. The sum of sine and cosine functions does not consistently equal 1 across all real values of x.

H. 7sin(7x) + 7cos(7x) = 14

This equation is similar to option F in that it can be ruled out by considering the range of sine and cosine functions. We can factor out the 7 from the left side of the equation to get 7[sin(7x) + cos(7x)] = 14. Dividing both sides by 7, we get sin(7x) + cos(7x) = 2. As we discussed in option F, the maximum possible value of sin(7x) + cos(7x) is 2. This occurs when sin(7x) and cos(7x) reach their maximum values in specific quadrants, which is possible but not consistent across all values of x. For the equation to hold true for all x, the sum should consistently be 2, which is not the case. The equation only holds true when both sine and cosine components are at their maximum, making it invalid for all real values of x.

J. sin²(7x) + cos²(7x) = 7

This equation is a direct violation of the Pythagorean identity. The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. Therefore, sin²(7x) + cos²(7x) must equal 1, not 7. This equation is definitively false for all real values of x. The violation of the fundamental Pythagorean Identity makes this option incorrect.

K. sin²(7x) + cos²(7x) = 1

This equation is the correct answer. It is a direct application of the Pythagorean identity, which, as we've discussed, holds true for all real values of x. Regardless of the value of x, the sum of the squares of sin(7x) and cos(7x) will always be 1. This identity is a fundamental trigonometric principle and is essential for solving a wide range of trigonometric problems. This equation is a direct representation of the Pythagorean Identity and hence, holds true for all real values of x.

The Power of Trigonometric Identities

This problem highlights the power and importance of trigonometric identities in simplifying and solving equations. The Pythagorean identity, in particular, is a cornerstone of trigonometry and is used extensively in various mathematical and scientific applications. By understanding and applying trigonometric identities, we can efficiently analyze and solve complex problems involving trigonometric functions. The correct answer, sin²(7x) + cos²(7x) = 1, underscores the significance of this identity.

Conclusion

In conclusion, for all real values of x, the equation sin²(7x) + cos²(7x) = 1 (option K) is the only one that holds true. This is a direct result of the Pythagorean identity, a fundamental principle in trigonometry. The other equations were shown to be false by considering the range of sine and cosine functions and the Pythagorean identity. This exercise demonstrates the importance of understanding and applying trigonometric identities to solve mathematical problems effectively. By mastering these concepts, you can confidently tackle a wide range of trigonometric challenges and deepen your understanding of the mathematical world.

#repair-input-keyword: Which equation is true for all real values of x? F. sin(7x) + cos(7x) = 7 G. sin(7x) + cos(7x) = 1 H. 7sin(7x) + 7cos(7x) = 14 J. sin²(7x) + cos²(7x) = 7 K. sin²(7x) + cos²(7x) = 1