Finding Sin(θ) In Quadrant III A Trigonometric Exploration
In the realm of trigonometry, understanding the relationships between angles and sides of triangles is paramount. The trigonometric functions, sine, cosine, and tangent, serve as the bedrock for these relationships. This article embarks on a journey to unravel the intricacies of finding the sine of an angle, specifically when the tangent of the angle and the quadrant in which it resides are known. We'll dive deep into the concepts, employing the given equation $ an ( heta)=\frac{5 \sqrt{11}}{11}$, and the knowledge that $\theta$ lies in the third quadrant, to accurately determine the value of $\sin (\theta)$.
The tangent function, mathematically represented as $ an(\theta)$, stands as a fundamental trigonometric ratio. It embodies the relationship between the opposite and adjacent sides of a right-angled triangle concerning a particular angle. In essence, it is the ratio of the sine of the angle to its cosine. When we're presented with $ an (\theta)=\frac{5 \sqrt{11}}{11}$, it's akin to holding a vital clue in a trigonometric puzzle. This value signifies that, in a right-angled triangle with angle $\theta$, the ratio of the side opposite to $ heta$ to the side adjacent to $\theta$ is $\frac{5 \sqrt{11}}{11}$. This piece of information is invaluable as it sets the stage for us to deduce the sine and cosine of the angle, but before we jump to conclusions, we need to consider the quadrant in which our angle $\theta$ resides.
The coordinate plane is segmented into four quadrants, each holding unique properties that dictate the signs of trigonometric functions. The third quadrant, our designated location for $\theta$, is a domain where both the x and y coordinates are negative. This is a critical detail because it directly influences the signs of the trigonometric functions. In the third quadrant, the sine and cosine functions are both negative, while the tangent function, being the ratio of sine to cosine, is positive. This understanding is crucial because, despite knowing the value of $ an(\theta)$, we must consider the sign conventions of the third quadrant to accurately determine $\sin (\theta)$. The positivity of $ an(\theta)$ in the third quadrant aligns with our given equation, confirming that our angle indeed lies within this region. This step is a vivid example of how trigonometry isn't just about formulas; it's about understanding the underlying geometry and how it impacts the signs and values of trigonometric functions.
To bridge the gap between the tangent and sine functions, we summon a powerful tool the Pythagorean trigonometric identity, $1 + \tan^2(\theta) = \sec^2(\theta)$. This identity is a cornerstone in trigonometry, linking tangent and secant, which in turn is related to cosine. By substituting our given value of $ an(\theta)$ into this identity, we can embark on a journey to find $\sec(\theta)$, and subsequently, $\cos(\theta)$. Let’s meticulously walk through the process. First, we square the tangent value: $(\frac5 \sqrt{11}}{11})^2 = \frac{25 \cdot 11}{121} = \frac{25}{11}$. Now, we plug this into our identity{11} = \sec^2(\theta)$. Simplifying, we get $\sec^2(\theta) = \frac{36}{11}$. Taking the square root of both sides, we find $|\sec(\theta)| = \frac{6}{\sqrt{11}}$. However, since $\theta$ is in the third quadrant, where cosine is negative, secant (which is the reciprocal of cosine) is also negative. Therefore, $\sec(\theta) = -\frac{6}{\sqrt{11}}$. From here, we can find $\cos(\theta)$ by taking the reciprocal of $\sec(\theta)$, giving us $\cos(\theta) = -\frac{\sqrt{11}}{6}$. This step exemplifies how trigonometric identities serve as navigational tools, guiding us through the landscape of trigonometric functions and enabling us to unearth hidden relationships.
With cosine now in our grasp, we pivot towards finding sine. The quintessential Pythagorean identity, $\sin^2(\theta) + \cos^2(\theta) = 1$, comes to our aid. This identity is a fundamental truth in trigonometry, representing the relationship between sine and cosine as sides of a unit circle. Substituting our calculated value of $\cos(\theta)$ into this identity, we set the stage for unveiling $\sin(\theta)$. We have $\sin^2(\theta) + (-\frac{\sqrt{11}}{6})^2 = 1$, which simplifies to $\sin^2(\theta) + \frac{11}{36} = 1$. Subtracting $\frac{11}{36}$ from both sides, we get $\sin^2(\theta) = \frac{25}{36}$. Taking the square root, we find $|\sin(\theta)| = \frac{5}{6}$. Now, recalling that $\theta$ resides in the third quadrant, where sine is negative, we conclude that $\sin(\theta) = -\frac{5}{6}$. This final step showcases the power of combining trigonometric identities with quadrant awareness to pinpoint the exact value of a trigonometric function. The journey from tangent to sine is a testament to the interconnectedness of trigonometric functions and the beauty of mathematical deduction.
In our exploration of trigonometric functions, we've successfully navigated the path from a given tangent value and quadrant location to the precise value of sine. This journey underscored the importance of understanding trigonometric ratios, quadrant rules, and the strategic application of Pythagorean identities. The value of $\sin(\theta)$, determined to be $-\frac{5}{6}$, stands as a testament to the power of trigonometric principles. This exercise not only reinforces the methodology for solving such problems but also deepens our appreciation for the elegance and interconnectedness of trigonometric concepts. As we conclude, remember that each trigonometric problem is a puzzle, and the tools we've discussed are the keys to unlocking their solutions.
The final answer is -\frac{5}{6}.