Finding Sine From Tangent In Quadrant II A Step By Step Solution

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In the realm of trigonometry, understanding the relationships between trigonometric functions and their values in different quadrants is crucial. This article delves into a specific problem that exemplifies this concept: given the tangent of an angle in the second quadrant, we aim to determine the sine of that angle. This task requires a firm grasp of trigonometric identities, quadrant rules, and algebraic manipulation. Let's embark on this mathematical journey together.

Understanding the Problem

Our primary goal is to determine the value of $\sin(\theta)$ given that $\tan(\theta) = -\frac5}{3}$ and $ heta$ lies in the second quadrant. This seemingly simple problem encapsulates several key trigonometric concepts. First, we need to recall the fundamental trigonometric identity that connects tangent, sine, and cosine $ an(\theta) = \frac{\sin(\theta){\cos(\theta)}$. Second, we must remember the signs of trigonometric functions in different quadrants. In the second quadrant, sine is positive, cosine is negative, and consequently, tangent is negative. These fundamental principles will guide our solution.

The negative value of the tangent immediately tells us that the angle cannot be in the first quadrant, where all trigonometric functions are positive. It also rules out the third quadrant, where both sine and cosine are negative, making the tangent positive. This leaves us with the second and fourth quadrants. The problem explicitly states that $\theta$ is in the second quadrant, which means that sine will be positive, and cosine will be negative. This crucial piece of information helps us to narrow down the possible solutions and ensures that our final answer aligns with the given conditions. The combination of trigonometric identities and quadrant rules forms the backbone of solving such problems, allowing us to navigate the relationships between different trigonometric functions with confidence.

Utilizing Trigonometric Identities

To solve this problem effectively, we must leverage the power of trigonometric identities. The identity that bridges the gap between tangent and sine is the Pythagorean identity: $1 + \tan^2(\theta) = \sec^2(\theta)$. This identity is a cornerstone of trigonometry, derived directly from the Pythagorean theorem applied to the unit circle. By substituting the given value of $\tan(\theta)$ into this identity, we can find the value of $\sec^2(\theta)$, and subsequently, $\cos^2(\theta)$. This is a crucial step because it allows us to transition from the known tangent value to a related trigonometric function, paving the way for finding the sine. The manipulation of trigonometric identities is a fundamental skill in trigonometry, enabling us to express trigonometric functions in different forms and solve a wide range of problems.

By substituting $ an(\theta) = -\frac5}{3}$ into the identity, we get$1 + \left(-\frac{5{3}\right)^2 = \sec^2(\theta)$$1 + \frac{25}{9} = \sec^2(\theta)$$\frac{34}{9} = \sec^2(\theta)$

Now, we know that $\sec(\theta) = \frac1}{\cos(\theta)}$, so $\cos^2(\theta) = \frac{1}{\sec^2(\theta)}$. Therefore, $\cos^2(\theta) = \frac{9}{34}$. Taking the square root of both sides gives us $\cos(\theta) = \pm\frac{3}{\sqrt{34}}$. Since $ heta$ is in the second quadrant, where cosine is negative, we choose the negative root $\cos(\theta) = -\frac{3{\sqrt{34}}$. This step highlights the importance of considering the quadrant when determining the sign of trigonometric functions. The correct sign is essential for arriving at the accurate solution. Having found the cosine, we are now well-positioned to determine the sine using another fundamental identity.

Finding Sine using the Pythagorean Identity

With the value of cosine determined, we can now employ the fundamental Pythagorean identity: $\sin^2(\theta) + \cos^2(\theta) = 1$. This identity is a cornerstone of trigonometry, directly derived from the Pythagorean theorem applied to the unit circle. It provides a powerful relationship between sine and cosine, allowing us to find one if we know the other. By substituting the value of $\cos(\theta)$ we found in the previous step, we can solve for $\sin^2(\theta)$, and subsequently, $\sin(\theta)$. This step demonstrates the interconnectedness of trigonometric functions and the utility of Pythagorean identities in solving trigonometric problems.

Substituting $\cos(\theta) = -\frac{3}{\sqrt{34}}$ into the Pythagorean identity, we get:

sin⁡2(θ)+(−334)2=1\sin^2(\theta) + \left(-\frac{3}{\sqrt{34}}\right)^2 = 1

sin⁡2(θ)+934=1\sin^2(\theta) + \frac{9}{34} = 1

sin⁡2(θ)=1−934\sin^2(\theta) = 1 - \frac{9}{34}

sin⁡2(θ)=2534\sin^2(\theta) = \frac{25}{34}

Taking the square root of both sides gives us $\sin(\theta) = \pm\frac5}{\sqrt{34}}$. Since $ heta$ is in the second quadrant, where sine is positive, we choose the positive root $\sin(\theta) = \frac{5{\sqrt{34}}$. To rationalize the denominator, we multiply both the numerator and denominator by $\sqrt{34}$, resulting in $\sin(\theta) = \frac{5\sqrt{34}}{34}$. This final step reinforces the importance of considering the quadrant when determining the sign of the trigonometric function and demonstrates the process of simplifying the result to its most standard form. Our journey through trigonometric identities and quadrant rules has led us to the solution.

The Solution

Therefore, given that $\tan(\theta) = -\frac{5}{3}$ and $ heta$ is in the second quadrant, the value of $\sin(\theta)$ is $\frac{5\sqrt{34}}{34}$. This solution highlights the interconnectedness of trigonometric functions and the importance of considering quadrant rules when solving trigonometric problems. By understanding these concepts, we can confidently navigate a wide range of trigonometric challenges.

Key Takeaways

This problem serves as a valuable illustration of several key concepts in trigonometry:

  • Trigonometric Identities: Mastering trigonometric identities, such as the Pythagorean identities and the relationship between tangent, sine, and cosine, is crucial for solving trigonometric problems.
  • Quadrant Rules: Understanding the signs of trigonometric functions in different quadrants is essential for determining the correct solution.
  • Algebraic Manipulation: Proficiency in algebraic manipulation is necessary for solving equations and simplifying expressions.
  • Interconnectedness: Trigonometric functions are interconnected, and solving for one function often involves using information about other functions.

By internalizing these takeaways, students can build a strong foundation in trigonometry and tackle more complex problems with confidence.

Conclusion

Determining the sine of an angle given its tangent and quadrant requires a solid understanding of trigonometric identities and quadrant rules. This problem exemplifies the power of these tools in navigating trigonometric relationships. By systematically applying these concepts, we successfully found the value of $\sin(\theta)$. This exercise underscores the importance of a holistic approach to trigonometry, where identities, quadrant rules, and algebraic skills work in harmony to unlock solutions. As we conclude this exploration, let us remember that the journey through trigonometry is not just about finding answers but about deepening our understanding of the beautiful relationships that govern the world of angles and functions.

Final Answer: The final answer is $\boxed{\frac{5 \sqrt{34}}{34}}$