Finding Tan Θ In Quadrant II A Step By Step Solution

In this comprehensive article, we will delve into the process of finding the value of tan(θ) when given the value of sin(θ) and the quadrant in which the angle θ lies. Specifically, we will address the equation sin(θ) = ${\frac{\sqrt{37}}{37}}$ and the condition that θ is an angle in quadrant II. This problem combines trigonometric identities, understanding of quadrants, and algebraic manipulation. Our goal is to provide a step-by-step solution that not only answers the question but also enhances your understanding of trigonometric concepts. This article is tailored for students, educators, and anyone interested in mathematics, particularly trigonometry. We will break down each step, ensuring clarity and providing additional insights to make the learning experience as enriching as possible. Whether you are preparing for an exam or simply expanding your knowledge, this guide will offer the necessary tools and understanding.

The core of our problem lies in determining the value of tan(θ) given that sin(θ) = ${\frac{\sqrt{37}}{37}}$ and θ is located in the second quadrant. This requires us to use the fundamental trigonometric identities and understand the sign conventions for trigonometric functions in different quadrants. The sine function, which is the ratio of the opposite side to the hypotenuse in a right-angled triangle, is given. However, to find the tangent function, which is the ratio of the opposite side to the adjacent side, we need to find the adjacent side. This involves using the Pythagorean identity, which relates sine and cosine. The added complexity of θ being in the second quadrant is crucial because it affects the sign of the cosine and, consequently, the tangent function. In quadrant II, the sine function is positive, while the cosine function is negative. This understanding is vital in arriving at the correct answer. We will meticulously walk through each step, ensuring that the reasoning behind every calculation is clear. This approach will not only help solve this particular problem but also build a solid foundation for tackling similar problems in trigonometry.

To effectively solve trigonometric problems, it’s crucial to deeply understand the definitions of trigonometric functions and how their signs vary across the four quadrants. Trigonometric functions, namely sine (sin), cosine (cos), and tangent (tan), are defined based on the ratios of sides in a right-angled triangle. Sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side. Understanding these definitions is the foundation for solving more complex problems. The unit circle provides a visual representation of these functions for all angles. Each quadrant represents a 90-degree section of the circle, and the signs of the trigonometric functions change depending on the quadrant. In the first quadrant (0° to 90°), all trigonometric functions are positive. In the second quadrant (90° to 180°), sine is positive, while cosine and tangent are negative. This is because the y-coordinate (representing sine) is positive, but the x-coordinate (representing cosine) is negative, and tangent is the ratio of y to x. In the third quadrant (180° to 270°), tangent is positive, while sine and cosine are negative. Finally, in the fourth quadrant (270° to 360°), cosine is positive, while sine and tangent are negative. This quadrant-specific behavior is essential for determining the correct sign of the trigonometric functions in various problems. By internalizing these principles, you can confidently approach a wide range of trigonometric questions and avoid common mistakes related to signs and function values.

The Pythagorean identity is a cornerstone in trigonometry, playing a crucial role in relating sine and cosine. This identity, expressed as sin²(θ) + cos²(θ) = 1, is derived from the Pythagorean theorem applied to the unit circle. It allows us to find the value of cosine when sine is known, or vice versa. In our problem, we are given sin(θ) = ${\frac{\sqrt{37}}{37}}$. To find cos(θ), we substitute the value of sin(θ) into the Pythagorean identity. This gives us ${(\frac{\sqrt{37}}{37}}$)² + cos²(θ) = 1. Simplifying this, we get ${\frac{37}{37^2}}$ + cos²(θ) = 1, which further simplifies to ${\frac{1}{37}}$ + cos²(θ) = 1. Now, we solve for cos²(θ) by subtracting ${\frac{1}{37}}$ from 1, resulting in cos²(θ) = ${\frac{36}{37}}$. Taking the square root of both sides, we find cos(θ) = ±${\frac{6}{\sqrt{37}}}$ or ±${\frac{6\sqrt{37}}{37}}$ after rationalizing the denominator. Since we know that θ is in the second quadrant, where cosine is negative, we choose the negative root. Therefore, cos(θ) = -${\frac{6\sqrt{37}}{37}}$. This step is vital because it provides us with the cosine value, which is essential for calculating the tangent. By mastering the use of the Pythagorean identity, you can navigate through many trigonometric problems with greater ease and precision. This fundamental relationship between sine and cosine is a powerful tool in trigonometric analysis and problem-solving.

Determining the sign of cos(θ) in Quadrant II is a critical step in solving our problem accurately. As previously discussed, the unit circle and the properties of quadrants dictate the signs of trigonometric functions. In the second quadrant, which spans angles between 90° and 180°, the x-coordinate is negative, and the y-coordinate is positive. Since cosine corresponds to the x-coordinate and sine corresponds to the y-coordinate, cosine is negative in this quadrant, while sine is positive. This understanding directly influences our choice of the root when solving for cos(θ) using the Pythagorean identity. We obtained cos²(θ) = ${\frac{36}{37}}$, which gives us two possible values for cos(θ): +${\frac{6\sqrt{37}}{37}}$ and -${\frac{6\sqrt{37}}{37}}$. However, given that θ is in the second quadrant, we must choose the negative value. Therefore, cos(θ) = -${\frac{6\sqrt{37}}{37}}$. This choice is not arbitrary; it is a direct consequence of the quadrant in which the angle lies. Neglecting the sign would lead to an incorrect final answer. This step highlights the importance of not just performing calculations but also understanding the underlying principles that govern trigonometric functions. By consistently applying the rules of quadrants, you can avoid common errors and build a more robust understanding of trigonometry.

Calculating tan(θ) is the final step in solving our problem. The tangent function is defined as the ratio of sine to cosine, that is, tan(θ) = ${\frac{sin(θ)}{cos(θ)}}$. We have already determined that sin(θ) = ${\frac{\sqrt{37}}{37}}$ and cos(θ) = -${\frac{6\sqrt{37}}{37}}$. Now, we can substitute these values into the formula for tan(θ). This gives us tan(θ) = ${\frac{\frac{\sqrt{37}}{37}}{-\frac{6\sqrt{37}}{37}}}$. To simplify this complex fraction, we multiply the numerator by the reciprocal of the denominator: tan(θ) = ${\frac{\sqrt{37}}{37}}$ * ${\frac{37}{-6\sqrt{37}}}$. We can cancel out the 37 in the numerator and denominator, as well as the ${\sqrt{37}}$. This leaves us with tan(θ) = ${\frac{1}{-6}}$, which simplifies to tan(θ) = -${\frac{1}{6}}$. The negative sign is consistent with the fact that tangent is negative in the second quadrant. This final calculation brings together all the previous steps, demonstrating how a solid understanding of trigonometric identities and quadrant rules leads to a correct solution. The ability to calculate the tangent function given sine and the quadrant is a fundamental skill in trigonometry. By following this methodical approach, you can confidently solve similar problems and deepen your understanding of trigonometric relationships.

After meticulously working through each step, we have arrived at the final answer. Given that sin(θ) = ${\frac{\sqrt{37}}{37}}$ and θ is an angle in the second quadrant, we have calculated the value of tan(θ). We used the Pythagorean identity to find cos(θ), taking into account that cosine is negative in the second quadrant. We then used the definition of tangent as the ratio of sine to cosine to find tan(θ). Our calculations led us to the result tan(θ) = -${\frac{1}{6}}$. This answer is consistent with the properties of trigonometric functions in the second quadrant, where sine is positive, cosine is negative, and tangent is negative. The negative sign is a crucial part of the solution and reflects our understanding of quadrant-specific behavior. This problem demonstrates the importance of not just knowing the trigonometric identities but also understanding how to apply them in conjunction with the rules of quadrants. The ability to solve such problems is a testament to a strong foundation in trigonometry and mathematical reasoning. With this comprehensive solution, you should now have a clear understanding of how to approach similar trigonometric problems with confidence.