Solving Trigonometric Equations Finding Cos(θ) In Quadrant IV

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Delving into Trigonometric Equations: Finding Cos(θ) in Quadrant IV

In the captivating realm of trigonometry, we often encounter equations that challenge our understanding of angles and their relationships. One such equation presents itself as: sin⁡(θ)=−255\sin (\theta) = -\frac{2 \sqrt{5}}{5}. This equation, seemingly simple at first glance, holds a wealth of information about the angle θ and its trigonometric properties. To fully unravel its secrets, we must delve into the fundamental principles of trigonometry and carefully consider the given conditions.

The problem further specifies that θ lies in quadrant IV, a crucial piece of information that significantly narrows down the possibilities. Quadrant IV, nestled in the bottom-right corner of the unit circle, is characterized by positive cosine values and negative sine values. This is a direct consequence of the definitions of sine and cosine in terms of the coordinates of a point on the unit circle. In quadrant IV, the x-coordinate (representing cosine) is positive, while the y-coordinate (representing sine) is negative.

Given that sin⁡(θ)=−255\sin (\theta) = -\frac{2 \sqrt{5}}{5}, we can leverage the fundamental trigonometric identity sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1 to unveil the value of cos⁡(θ)\cos(\theta). This identity, a cornerstone of trigonometry, establishes an unbreakable relationship between the sine and cosine of any angle. By substituting the known value of sin⁡(θ)\sin(\theta) into this identity, we can embark on a journey to discover the elusive cos⁡(θ)\cos(\theta).

Substituting sin⁡(θ)=−255\sin (\theta) = -\frac{2 \sqrt{5}}{5} into the identity, we get:

(−255)2+cos⁡2(θ)=1\left(-\frac{2 \sqrt{5}}{5}\right)^2 + \cos^2(\theta) = 1

Simplifying the equation, we have:

2025+cos⁡2(θ)=1\frac{20}{25} + \cos^2(\theta) = 1

45+cos⁡2(θ)=1\frac{4}{5} + \cos^2(\theta) = 1

Isolating cos⁡2(θ)\cos^2(\theta), we obtain:

cos⁡2(θ)=1−45\cos^2(\theta) = 1 - \frac{4}{5}

cos⁡2(θ)=15\cos^2(\theta) = \frac{1}{5}

Now, to find cos⁡(θ)\cos(\theta), we take the square root of both sides:

cos⁡(θ)=±15\cos(\theta) = \pm \sqrt{\frac{1}{5}}

cos⁡(θ)=±55\cos(\theta) = \pm \frac{\sqrt{5}}{5}

At this juncture, we encounter two possible solutions: 55\frac{\sqrt{5}}{5} and −55-\frac{\sqrt{5}}{5}. However, the condition that θ lies in quadrant IV comes to our rescue. As we established earlier, cosine values are positive in quadrant IV. Therefore, we can confidently discard the negative solution and embrace the positive one.

Thus, the value of cos⁡(θ)\cos(\theta) is 55\frac{\sqrt{5}}{5}. This elegantly solves the trigonometric puzzle, revealing the cosine of the angle θ in quadrant IV.

Decoding Trigonometric Relationships: A Step-by-Step Solution

To solidify our understanding, let's recap the steps we took to arrive at the solution:

  1. We began by recognizing the given equation, sin⁡(θ)=−255\sin (\theta) = -\frac{2 \sqrt{5}}{5}, and acknowledging the crucial information that θ lies in quadrant IV.
  2. We recalled the fundamental trigonometric identity sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1, a cornerstone of trigonometric relationships.
  3. We skillfully substituted the known value of sin⁡(θ)\sin(\theta) into the identity, paving the way for us to isolate cos⁡2(θ)\cos^2(\theta).
  4. We simplified the equation and extracted the square root, unveiling two potential solutions for cos⁡(θ)\cos(\theta).
  5. We judiciously applied the quadrant condition, recognizing that cosine values are positive in quadrant IV, and discarded the negative solution.
  6. Finally, we triumphantly declared the value of cos⁡(θ)\cos(\theta) as 55\frac{\sqrt{5}}{5}, completing our trigonometric quest.

This step-by-step approach not only leads us to the correct answer but also reinforces our grasp of trigonometric principles. By understanding the interplay between sine, cosine, and quadrant conditions, we equip ourselves to tackle a wide array of trigonometric challenges.

The Significance of Quadrant IV: Unveiling the Sign Conventions

The quadrant condition plays a pivotal role in determining the signs of trigonometric functions. In quadrant IV, the cosine function reigns supreme with its positive values, while the sine function dwells in the negative realm. This sign convention stems from the geometric interpretation of trigonometric functions on the unit circle.

Imagine a point traversing the unit circle, its coordinates dictating the values of sine and cosine. In quadrant IV, the x-coordinate of this point is positive, corresponding to a positive cosine value. Conversely, the y-coordinate is negative, reflecting the negative sine value. This visual representation solidifies the understanding of sign conventions in different quadrants.

Understanding quadrant sign conventions is not merely a matter of rote memorization; it is a key to unlocking the deeper meaning of trigonometric functions. By grasping the geometric origins of these conventions, we can confidently navigate trigonometric problems and interpret the results in a meaningful way.

Mastering Trigonometric Identities: The Key to Unlocking Solutions

Trigonometric identities, such as sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1, serve as powerful tools in solving trigonometric equations. These identities establish fundamental relationships between trigonometric functions, allowing us to manipulate equations and isolate the desired variables.

The identity sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1 is arguably the most fundamental trigonometric identity. It stems directly from the Pythagorean theorem applied to the unit circle. By mastering this identity and its variations, we gain a significant advantage in solving trigonometric problems.

In this particular problem, the identity served as the bridge between the known value of sin⁡(θ)\sin(\theta) and the unknown value of cos⁡(θ)\cos(\theta). By substituting the known value and strategically manipulating the equation, we were able to unveil the solution. This exemplifies the power of trigonometric identities in simplifying complex problems.

Conclusion: Embracing the Beauty of Trigonometry

In conclusion, the problem of finding cos⁡(θ)\cos(\theta) given sin⁡(θ)=−255\sin (\theta) = -\frac{2 \sqrt{5}}{5} and the quadrant condition epitomizes the beauty and elegance of trigonometry. By skillfully applying trigonometric identities and considering quadrant conditions, we can unravel the secrets of angles and their relationships.

This journey through trigonometric equations underscores the importance of a solid foundation in trigonometric principles. By mastering trigonometric identities, understanding quadrant conventions, and embracing a step-by-step approach, we can confidently tackle a wide range of trigonometric challenges. Trigonometry, far from being a collection of formulas and rules, is a gateway to a deeper understanding of the mathematical world and its applications in various fields.

Therefore, the correct answer is A. 55\frac{\sqrt{5}}{5}