Trigonometric Identity And Sine Value In Second Quadrant A Comprehensive Explanation

Introduction

In the realm of trigonometry, the fundamental trigonometric identity sin² θ + cos² θ = 1 plays a pivotal role. This relationship holds true for all real values of θ. However, when combined with specific domain restrictions, such as π/2 < θ < π, we can delve deeper into the characteristics of trigonometric functions and their values within particular quadrants. This article aims to elucidate the interplay between this identity and the determination of sin θ within the specified domain, offering a comprehensive understanding for students and enthusiasts alike.

The Fundamental Trigonometric Identity

The cornerstone of trigonometric relationships is the Pythagorean identity: sin² θ + cos² θ = 1. This identity is derived directly from the Pythagorean theorem applied to the unit circle. A point on the unit circle can be represented as (cos θ, sin θ), where the radius of the circle is 1. The theorem states that in a right-angled triangle, the square of the hypotenuse (1 in this case) is equal to the sum of the squares of the other two sides (cos θ and sin θ). Hence, we arrive at the identity.

This identity is not merely a formula; it's a foundational principle that connects sine and cosine functions. It allows us to express one trigonometric function in terms of the other, given the value of one and the quadrant in which the angle lies. For example, if we know the value of cos θ, we can find sin θ using this identity, and vice versa. The sign of the trigonometric function (positive or negative) is determined by the quadrant in which the angle θ is situated. This is where the given restriction π/2 < θ < π becomes crucial, as it places θ in the second quadrant.

The Pythagorean identity serves as a bedrock for deriving numerous other trigonometric identities and formulas. It's used extensively in simplifying trigonometric expressions, solving equations, and proving other trigonometric relationships. Understanding this identity thoroughly is essential for mastering trigonometry and its applications in various fields, including physics, engineering, and computer graphics. The elegance of this identity lies in its simplicity and its far-reaching implications, making it an indispensable tool in the world of mathematics.

The Significance of the Domain Restriction: π/2 < θ < π

The domain restriction π/2 < θ < π is of paramount importance because it confines the angle θ to the second quadrant of the unit circle. In the unit circle, the quadrants are numbered counterclockwise, starting from the positive x-axis as the first quadrant. The second quadrant is where the x-coordinates are negative, and the y-coordinates are positive. This has direct implications for the signs of trigonometric functions.

In the second quadrant, the sine function (sin θ) is positive because it corresponds to the y-coordinate of a point on the unit circle. Conversely, the cosine function (cos θ) is negative in this quadrant as it corresponds to the x-coordinate. The tangent function (tan θ), which is the ratio of sin θ to cos θ, is therefore negative in the second quadrant. Understanding these sign conventions is crucial for accurately determining the values of trigonometric functions within specific domains.

This restriction allows us to narrow down the possible solutions when solving trigonometric equations or evaluating expressions. Without this constraint, there could be multiple angles that satisfy a given trigonometric equation due to the periodic nature of trigonometric functions. By limiting the domain, we can pinpoint the specific angle that meets the given conditions.

For instance, if we are given the value of cos θ and the restriction π/2 < θ < π, we can use the Pythagorean identity to find sin θ. Since we know that sin θ is positive in the second quadrant, we choose the positive square root when solving for sin θ. This illustrates how the domain restriction, combined with the trigonometric identity, enables us to determine the unique value of sin θ for the given condition. The interplay between domain restrictions and trigonometric identities is a fundamental concept in trigonometry, providing a framework for solving a wide range of problems.

Determining the Value of sin θ

Given the relationship sin² θ + cos² θ = 1 and the domain π/2 < θ < π, we can determine the value of sin θ if we know the value of cos θ within this interval. The process involves using the Pythagorean identity to express sin θ in terms of cos θ, and then considering the sign of sin θ in the second quadrant.

To begin, we rearrange the identity to isolate sin² θ: sin² θ = 1 - cos² θ. This equation shows that sin² θ is dependent on the value of cos² θ. Next, we take the square root of both sides to solve for sin θ: sin θ = ±√(1 - cos² θ). It is at this juncture that the domain restriction becomes crucial. In the second quadrant (π/2 < θ < π), the sine function is positive. Therefore, we choose the positive square root:

sin θ = √(1 - cos² θ)

This equation gives us the magnitude of sin θ. The sign is already determined by the quadrant. For example, if cos θ = -0.6, we can calculate sin θ as follows:

sin θ = √(1 - (-0.6)²) = √(1 - 0.36) = √0.64 = 0.8

This calculation illustrates how the Pythagorean identity and the domain restriction work in tandem to yield a unique positive value for sin θ. The domain restriction eliminates the ambiguity of the ± sign that arises from taking the square root, ensuring that we obtain the correct value for sin θ in the specified quadrant. This method is widely used in trigonometric problem-solving, emphasizing the importance of understanding both the identities and the quadrant rules.

Examples and Applications

To further illustrate the relationship between sin² θ + cos² θ = 1 and the value of sin θ in the second quadrant, let's consider a couple of examples. These examples will demonstrate how to apply the concepts discussed earlier to solve practical problems.

Example 1: Suppose we are given that cos θ = -√3/2 and π/2 < θ < π. Find the value of sin θ.

Using the identity sin² θ + cos² θ = 1, we have:

sin² θ = 1 - cos² θ

Substitute the given value of cos θ:

sin² θ = 1 - (-√3/2)² = 1 - 3/4 = 1/4

Taking the square root of both sides gives us:

sin θ = ±√(1/4) = ±1/2

Since θ is in the second quadrant, sin θ is positive. Therefore,

sin θ = 1/2

Example 2: Let's say sin θ = √2/2 and π/2 < θ < π. Determine the value of cos θ.

Starting with the identity sin² θ + cos² θ = 1:

cos² θ = 1 - sin² θ

Substitute the given value of sin θ:

cos² θ = 1 - (√2/2)² = 1 - 2/4 = 1 - 1/2 = 1/2

Taking the square root:

cos θ = ±√(1/2) = ±√2/2

In the second quadrant, cos θ is negative, so:

cos θ = -√2/2

These examples highlight the step-by-step process of using the Pythagorean identity and the quadrant rules to find the values of trigonometric functions. These principles are not just theoretical constructs; they have numerous real-world applications. In physics, they are used to analyze wave motion and projectile trajectories. Engineers use them in structural analysis and electrical circuit design. Even in computer graphics, these trigonometric concepts are essential for creating realistic 3D models and animations. The ability to apply these principles effectively is a valuable skill in many scientific and technical fields.

Conclusion

In conclusion, the relationship sin² θ + cos² θ = 1 is a cornerstone of trigonometry, and its application is greatly influenced by the domain restriction placed on the angle θ. When π/2 < θ < π, we are working within the second quadrant, where sin θ is positive and cos θ is negative. This knowledge, combined with the Pythagorean identity, allows us to determine the value of sin θ (or cos θ) when the other is known. Understanding these concepts is crucial for success in trigonometry and its many applications in science, engineering, and beyond. The examples provided serve to reinforce these principles and demonstrate their practical utility. The interplay between trigonometric identities and domain restrictions forms a powerful toolset for solving a wide range of problems, making it an essential area of study for anyone pursuing mathematics or related fields.