Trigonometric Functions In Quadrant II Sin Θ = 1/7

In the realm of trigonometry, understanding the relationships between trigonometric functions and the quadrants of the coordinate plane is crucial. This article delves into a specific scenario where the sine of an angle, denoted as sin θ, is given as 1/7, and the terminal side of the angle lies in Quadrant II. We will explore how to determine the exact values of the other trigonometric functions for this angle. This involves understanding the definitions of trigonometric functions, the Pythagorean identity, and the sign conventions in different quadrants. By meticulously applying these principles, we can unravel the values of cosine, tangent, cosecant, secant, and cotangent for the given angle θ. This exploration not only enhances our understanding of trigonometric functions but also provides a practical approach to solving trigonometric problems.

Determining Cosine (cos θ) in Quadrant II

Given that sin θ = 1/7, and the terminal side of θ is in Quadrant II, our initial task is to find the exact value of cos θ. The fundamental trigonometric identity that links sine and cosine is the Pythagorean identity: sin² θ + cos² θ = 1. This identity is a cornerstone of trigonometry, derived directly from the Pythagorean theorem applied to the unit circle. It allows us to relate the sine and cosine of any angle, making it an indispensable tool in solving trigonometric problems.

Substituting the given value of sin θ into the Pythagorean identity, we get (1/7)² + cos² θ = 1. Simplifying this equation, we have 1/49 + cos² θ = 1. To isolate cos² θ, we subtract 1/49 from both sides of the equation, resulting in cos² θ = 1 - 1/49. This simplifies to cos² θ = 48/49. Now, to find cos θ, we take the square root of both sides of the equation. However, it's crucial to remember that taking the square root yields both positive and negative solutions. Therefore, we have cos θ = ±√(48/49).

At this juncture, the quadrant information becomes paramount. Since the terminal side of θ lies in Quadrant II, we must consider the sign convention for cosine in this quadrant. In Quadrant II, the x-coordinates are negative, and cosine corresponds to the x-coordinate on the unit circle. Hence, cos θ must be negative in Quadrant II. Consequently, we choose the negative square root, giving us cos θ = -√(48/49). Simplifying the radical, we can express √48 as √(16 * 3), which is 4√3. The square root of 49 is 7. Therefore, cos θ = -4√3 / 7. This is the exact value of cosine for the given angle θ.

In summary, by leveraging the Pythagorean identity and the quadrant information, we have successfully determined the exact value of cos θ. The process involved substituting the known sine value into the Pythagorean identity, solving for cos² θ, and then taking the square root. The crucial step was then considering the quadrant in which the angle terminates to determine the correct sign for cosine. This methodical approach highlights the importance of understanding both trigonometric identities and quadrant sign conventions in solving trigonometric problems. The value cos θ = -4√3 / 7 is a precise representation of the cosine of the angle, taking into account its location in Quadrant II.

Determining the Remaining Trigonometric Function Values

With the values of sin θ and cos θ established, we can now calculate the remaining trigonometric functions: tangent (tan θ), cosecant (csc θ), secant (sec θ), and cotangent (cot θ). Each of these functions is defined in terms of sine and cosine, allowing us to systematically determine their values. This section will delve into the definitions of these functions, their relationships to sine and cosine, and the application of these relationships to find the exact values for our given scenario where sin θ = 1/7 and cos θ = -4√3 / 7, with θ in Quadrant II.

Tangent (tan θ)

The tangent function is defined as the ratio of sine to cosine: tan θ = sin θ / cos θ. Substituting the values we have, tan θ = (1/7) / (-4√3 / 7). To simplify this complex fraction, we multiply the numerator and the denominator by 7, which cancels out the denominators, giving us tan θ = 1 / (-4√3). To rationalize the denominator, we multiply both the numerator and the denominator by √3, resulting in tan θ = -√3 / (4 * 3), which simplifies to tan θ = -√3 / 12. This is the exact value of the tangent function for the given angle θ.

Cosecant (csc θ)

The cosecant function is the reciprocal of the sine function: csc θ = 1 / sin θ. Given sin θ = 1/7, the cosecant is simply the reciprocal of this value. Therefore, csc θ = 1 / (1/7) = 7. The exact value of the cosecant function is 7, which is a straightforward calculation due to the reciprocal relationship with sine.

Secant (sec θ)

The secant function is the reciprocal of the cosine function: sec θ = 1 / cos θ. We found that cos θ = -4√3 / 7, so sec θ = 1 / (-4√3 / 7). To simplify this, we take the reciprocal of the fraction, giving us sec θ = -7 / (4√3). To rationalize the denominator, we multiply both the numerator and the denominator by √3. This yields sec θ = -7√3 / (4 * 3), which simplifies to sec θ = -7√3 / 12. This is the exact value of the secant function for the given angle θ.

Cotangent (cot θ)

The cotangent function is the reciprocal of the tangent function: cot θ = 1 / tan θ. We calculated tan θ = -√3 / 12, so cot θ = 1 / (-√3 / 12). Taking the reciprocal, we get cot θ = -12 / √3. To rationalize the denominator, we multiply both the numerator and the denominator by √3. This gives us cot θ = -12√3 / 3, which simplifies to cot θ = -4√3. This is the exact value of the cotangent function for the given angle θ.

In summary, by utilizing the definitions and reciprocal relationships of the trigonometric functions, we have successfully determined the exact values of tan θ, csc θ, sec θ, and cot θ. Each calculation involved substituting the known values of sin θ and cos θ and simplifying the resulting expressions. Rationalizing denominators was a key step in expressing the values in their simplest forms. These calculations demonstrate the interconnectedness of the trigonometric functions and the importance of understanding their definitions and relationships.

Significance of Quadrant II in Determining Trigonometric Function Signs

Understanding the significance of Quadrant II is crucial when determining the signs of trigonometric functions. The coordinate plane is divided into four quadrants, each with distinct sign patterns for the x and y coordinates. These sign patterns directly influence the signs of trigonometric functions, as they are defined in terms of these coordinates on the unit circle. In Quadrant II, the x-coordinates are negative, and the y-coordinates are positive. This characteristic is the key to understanding why certain trigonometric functions are positive or negative in this quadrant.

Sign Conventions in Quadrant II

To fully grasp the significance of Quadrant II, let's examine the sign conventions for each trigonometric function: Sine (sin θ) is defined as the ratio of the y-coordinate to the radius (which is always positive). In Quadrant II, since the y-coordinate is positive, sin θ is also positive. This is consistent with our given condition that sin θ = 1/7.

Cosine (cos θ) is defined as the ratio of the x-coordinate to the radius. In Quadrant II, the x-coordinate is negative, making cos θ negative. This is why, in our previous calculations, we chose the negative square root when finding cos θ from the Pythagorean identity.

Tangent (tan θ) is the ratio of sin θ to cos θ. In Quadrant II, a positive sin θ divided by a negative cos θ results in a negative tan θ. This aligns with our calculated value of tan θ = -√3 / 12.

Cosecant (csc θ) is the reciprocal of sin θ. Since sin θ is positive in Quadrant II, csc θ is also positive. Our calculation of csc θ = 7 confirms this.

Secant (sec θ) is the reciprocal of cos θ. Because cos θ is negative in Quadrant II, sec θ is also negative. Our result of sec θ = -7√3 / 12 is consistent with this sign convention.

Cotangent (cot θ) is the reciprocal of tan θ. As tan θ is negative in Quadrant II, cot θ is also negative. Our calculated value of cot θ = -4√3 supports this.

The ASTC Rule

A helpful mnemonic to remember which trigonometric functions are positive in each quadrant is the ASTC rule: All, Sine, Tangent, Cosine. This acronym represents the functions that are positive in each quadrant, starting from Quadrant I and moving counterclockwise:

Quadrant I: All trigonometric functions are positive. Quadrant II: Sine (and its reciprocal, cosecant) are positive. Quadrant III: Tangent (and its reciprocal, cotangent) are positive. Quadrant IV: Cosine (and its reciprocal, secant) are positive.

The ASTC rule provides a quick reference for determining the signs of trigonometric functions in different quadrants. In the context of our problem, the fact that θ is in Quadrant II immediately tells us that sine and cosecant are positive, while cosine, tangent, secant, and cotangent are negative. This knowledge is invaluable when solving trigonometric equations and problems.

In summary, the significance of Quadrant II in determining the signs of trigonometric functions cannot be overstated. The sign conventions in each quadrant are a direct consequence of the signs of the x and y coordinates, which in turn dictate the signs of the trigonometric ratios. Understanding these conventions and utilizing tools like the ASTC rule are essential for accurate trigonometric calculations and problem-solving. In our specific example, knowing that θ is in Quadrant II guided us in choosing the correct signs for cosine and the other related trigonometric functions.

Conclusion

In conclusion, determining the exact values of trigonometric functions when given one function value and the quadrant of the terminal side requires a thorough understanding of trigonometric identities, definitions, and sign conventions. In this article, we explored the scenario where sin θ = 1/7 and the terminal side of θ lies in Quadrant II. We successfully calculated the exact values of cos θ, tan θ, csc θ, sec θ, and cot θ by applying the Pythagorean identity, the definitions of trigonometric functions, and the ASTC rule for quadrant signs. The process involved substituting the known sine value into the Pythagorean identity to find cosine, considering the negative sign for cosine in Quadrant II. Subsequently, we used the definitions of tangent, cosecant, secant, and cotangent in terms of sine and cosine to compute their respective values, ensuring that the signs were consistent with Quadrant II. Understanding the significance of Quadrant II in determining the signs of trigonometric functions is paramount. The negative x-coordinates and positive y-coordinates in this quadrant dictate the signs of cosine, tangent, secant, and cotangent, while sine and cosecant remain positive. This knowledge, combined with the ASTC rule, provides a robust framework for solving trigonometric problems. This methodical approach not only yields accurate results but also reinforces the fundamental principles of trigonometry. By mastering these principles, one can confidently tackle a wide range of trigonometric challenges and gain a deeper appreciation for the elegance and interconnectedness of these mathematical concepts.