Trigonometric Equations Finding The Identity True For All Real X

In the realm of trigonometry, certain fundamental identities hold the key to unraveling complex equations. When dealing with trigonometric functions like sine and cosine, it's crucial to remember the core relationships that govern their behavior. In this comprehensive exploration, we will dissect the given equations and leverage the fundamental trigonometric identity to pinpoint the equation that remains true for all real values of x. Our journey will delve into the properties of sine and cosine, their bounded nature, and how the Pythagorean identity plays a pivotal role in simplifying trigonometric expressions. By the end of this analysis, you'll have a solid understanding of why one equation stands out as the correct answer and the reasoning behind it.

Analyzing the Equations

Let's scrutinize each equation provided, keeping in mind the inherent properties of sine and cosine functions. The sine and cosine functions are bounded, meaning their values always fall within the range of -1 to 1, inclusive. This crucial fact will help us eliminate several options quickly.

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

The first equation, $\sin (7 x)+\cos (7 x)=7$, immediately raises a red flag. As we know, the maximum value of both sine and cosine is 1. Therefore, even in the most optimistic scenario where both $\sin (7x)$ and $\cos (7x)$ simultaneously attain their maximum value of 1, their sum would only be 2. This is significantly less than 7. Consequently, this equation cannot hold true for all real values of x. The bounded nature of sine and cosine makes this equation inherently false.

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

Next, we examine the equation $\sin (7 x)+\cos (7 x)=1$. While it's conceivable that certain values of x might satisfy this equation, it's not universally true for all real values of x. To illustrate, consider the case where $x = 0$. In this instance, $\sin (7 * 0) = \sin (0) = 0$ and $\cos (7 * 0) = \cos (0) = 1$. The equation holds true. However, if we take a different value, such as $x = \frac{\pi}{14}$, then $\sin (7 * \frac{\pi}{14}) = \sin (\frac{\pi}{2}) = 1$ and $\cos (7 * \frac{\pi}{14}) = \cos (\frac{\pi}{2}) = 0$. Again, the equation holds true. But, it is crucial to realize that this equation doesn't hold true for every single x. For instance, there are other values of x where the sum does not equal 1. Therefore, this equation is not universally valid.

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

The third equation, $7 \sin (7 x)+7 \cos (7 x)=14$, can be simplified by dividing both sides by 7, resulting in $\sin (7 x)+\cos (7 x)=2$. Similar to the first equation, this equation is also flawed. As we've established, the maximum sum of $\sin (7x)$ and $\cos (7x)$ is 2 (when both functions are at their maximum, which never happens simultaneously), but reaching 2 requires very specific conditions. It's impossible for their sum to consistently equal 2 for all real values of x. Thus, this equation is incorrect.

J. $\sin ^2(7 x)+\cos ^2(7 x)=7$

Now, let's consider the equation $\sin ^2(7 x)+\cos ^2(7 x)=7$. This equation clashes directly with the fundamental trigonometric identity, also known as the Pythagorean identity. This identity states that for any angle θ, $\sin^2(θ) + \cos^2(θ) = 1$. Regardless of the value of x, $\sin ^2(7 x)+\cos ^2(7 x)$ will always equal 1. It will never equal 7. Hence, this equation is definitively false.

K. $\sin ^2(7 x)+\cos ^2(7 x)=1$

Finally, we arrive at the equation $\sin ^2(7 x)+\cos ^2(7 x)=1$. This equation is the embodiment of the Pythagorean identity. As mentioned earlier, the Pythagorean identity is a cornerstone of trigonometry, asserting that for any angle, the sum of the squares of its sine and cosine is always 1. There are no exceptions to this rule. This equation holds true for all real values of x, making it the correct answer.

The Pythagorean Identity: A Deeper Dive

The Pythagorean identity, $\sin^2(θ) + \cos^2(θ) = 1$, is not just a formula; it's a fundamental truth derived from the very definition of sine and cosine in the context of the unit circle. Consider a unit circle (a circle with a radius of 1) centered at the origin of a coordinate plane. For any point on this circle, the coordinates can be represented as (cos θ, sin θ), where θ is the angle formed between the positive x-axis and the line segment connecting the origin to the point. Applying the Pythagorean theorem to the right triangle formed by the x-coordinate, y-coordinate, and the radius (which is 1) directly yields the identity.

This identity is not merely a mathematical curiosity; it's a powerful tool that simplifies trigonometric expressions and solves equations. It allows us to express sine in terms of cosine, and vice versa, making it an indispensable part of trigonometric manipulations. In more advanced contexts, the Pythagorean identity serves as a building block for deriving other trigonometric identities and tackling complex problems in calculus and physics. Its consistent validity across all real values of the angle makes it a reliable and essential tool in mathematics.

Why Other Options Fail

To solidify our understanding, let's reiterate why the other options are incorrect:

• Option F: $\sin (7 x)+\cos (7 x)=7$ fails because the maximum sum of sine and cosine is significantly less than 7 due to their bounded nature.
• Option G: $\sin (7 x)+\cos (7 x)=1$ is not universally true; it only holds for specific values of x, not for all real values.
• Option H: $7 \sin (7 x)+7 \cos (7 x)=14$ simplifies to $\sin (7 x)+\cos (7 x)=2$, which is also impossible for all real values of x due to the bounds of sine and cosine.
• Option J: $\sin ^2(7 x)+\cos ^2(7 x)=7$ directly contradicts the Pythagorean identity, which dictates that the sum of the squares of sine and cosine is always 1.

Only option K, $\sin ^2(7 x)+\cos ^2(7 x)=1$, aligns perfectly with the universally true Pythagorean identity.

In conclusion, the equation that holds true for all real values of x is K. $\sin ^2(7 x)+\cos ^2(7 x)=1$. This is a direct consequence of the fundamental trigonometric identity, the Pythagorean identity, which is a cornerstone of trigonometry. Understanding this identity and the bounded nature of sine and cosine is crucial for solving trigonometric problems and simplifying expressions. The other equations fail because they either violate the bounded nature of sine and cosine or contradict the Pythagorean identity. Therefore, option K stands as the unequivocally correct answer, highlighting the importance of foundational trigonometric principles in mathematical problem-solving.