Calculating Sin(2θ) When Cos(θ) Is Known Trigonometry Exploration
In the realm of trigonometry, the double-angle formulas serve as powerful tools for unraveling the relationships between trigonometric functions of an angle and its double. Among these formulas, the sine double-angle formula, sin(2θ) = 2sin(θ)cos(θ), stands out as a cornerstone for solving a myriad of trigonometric problems. This article delves into the intricacies of calculating sin(2θ) when cos(θ) is given, particularly when the angle θ resides in the first quadrant. We will embark on a step-by-step journey, starting with the fundamental concepts and culminating in the final solution, while also highlighting the significance of quadrant information in determining the sign of trigonometric functions.
Decoding the Sine Double-Angle Formula
The sine double-angle formula, sin(2θ) = 2sin(θ)cos(θ), elegantly expresses the sine of twice an angle (2θ) in terms of the sine and cosine of the original angle (θ). This formula stems from the more general angle addition formula for sine, which states that sin(α + β) = sin(α)cos(β) + cos(α)sin(β). By setting α = β = θ, we arrive at the sine double-angle formula. This formula is not just a mathematical curiosity; it is a practical tool used extensively in various fields, including physics, engineering, and computer graphics, to analyze periodic phenomena, wave motion, and signal processing.
Understanding the derivation of the sine double-angle formula provides a deeper appreciation of its validity and applicability. It reinforces the interconnectedness of trigonometric identities and fosters a more intuitive understanding of trigonometric relationships. Moreover, this understanding allows for the formula to be adapted and applied to more complex scenarios, making it an indispensable asset in any mathematical toolkit.
The ability to manipulate and apply trigonometric identities like the sine double-angle formula is a critical skill for anyone working with periodic functions or oscillatory systems. It allows for the simplification of complex expressions, the solution of trigonometric equations, and the analysis of wave phenomena. This skill is particularly relevant in fields like electrical engineering, where sinusoidal waveforms are ubiquitous, and in physics, where the study of oscillations and waves is paramount.
Unraveling the Given Information: cos(θ) = 3√10/10 and Quadrant I
We are given that cos(θ) = 3√10/10, a positive value, and that θ lies in Quadrant I. This seemingly simple piece of information holds the key to unlocking the value of sin(2θ). The fact that cos(θ) is positive tells us that the adjacent side of the right triangle formed by θ is positive relative to the hypotenuse. The hypotenuse is always considered positive, so a positive cosine value implies a positive adjacent side. Furthermore, the specification that θ is in Quadrant I is crucial because it dictates the signs of all trigonometric functions. In Quadrant I, both sine and cosine are positive, ensuring that our calculations align with the geometric interpretation of trigonometric functions.
The quadrant information is not merely a side note; it is a fundamental aspect of trigonometric problem-solving. Each quadrant has a unique combination of positive and negative trigonometric function values. For example, in Quadrant II, sine is positive while cosine and tangent are negative. In Quadrant III, tangent is positive, while sine and cosine are negative. And in Quadrant IV, cosine is positive, while sine and tangent are negative. Understanding these sign conventions is essential for accurately determining the values of trigonometric functions and avoiding errors in calculations.
The value of cos(θ) itself provides us with valuable information about the shape of the right triangle associated with the angle θ. Since cos(θ) is the ratio of the adjacent side to the hypotenuse, we know that the adjacent side is 3√10 units long and the hypotenuse is 10 units long (or proportional to these values). This information will be crucial in determining the length of the opposite side, which is necessary for calculating sin(θ).
The Pythagorean Identity: Your Gateway to sin(θ)
The Pythagorean identity, sin²(θ) + cos²(θ) = 1, is a cornerstone of trigonometry, providing a fundamental relationship between the sine and cosine of an angle. This identity is a direct consequence of the Pythagorean theorem applied to the unit circle, where the radius is 1. By substituting the given value of cos(θ) into this identity, we can solve for sin(θ), thereby bridging the gap between the known cosine value and the desired sine value. This identity is not just a formula; it's a powerful tool that connects the geometric representation of trigonometric functions with their algebraic expressions.
The Pythagorean identity is a versatile tool that extends far beyond simple calculations. It is used in a wide range of applications, including simplifying trigonometric expressions, solving trigonometric equations, and deriving other trigonometric identities. Its importance stems from its fundamental connection to the geometry of the unit circle and the Pythagorean theorem, which are cornerstones of Euclidean geometry.
To find sin(θ), we substitute cos(θ) = 3√10/10 into the Pythagorean identity: sin²(θ) + (3√10/10)² = 1. This equation allows us to isolate sin²(θ) and subsequently find sin(θ). However, it is crucial to remember that taking the square root can yield both positive and negative solutions. This is where the quadrant information becomes invaluable. Since θ is in Quadrant I, we know that sin(θ) must be positive. This eliminates the negative solution and ensures that we arrive at the correct value for sin(θ).
Calculating sin(θ): A Step-by-Step Approach
Now, let's embark on the calculation of sin(θ) using the Pythagorean identity. We begin by substituting the given value of cos(θ) into the identity: sin²(θ) + (3√10/10)² = 1. Squaring 3√10/10 gives us 9 * 10 / 100 = 90/100 = 9/10. So, our equation becomes sin²(θ) + 9/10 = 1. Subtracting 9/10 from both sides, we get sin²(θ) = 1 - 9/10 = 1/10. Taking the square root of both sides, we obtain sin(θ) = ±√(1/10). Since θ is in Quadrant I, we choose the positive root, giving us sin(θ) = √(1/10).
To rationalize the denominator, we multiply both the numerator and denominator by √10, resulting in sin(θ) = √10 / 10. This is the exact value of sin(θ), which we will use in the next step to calculate sin(2θ). The process of rationalizing the denominator is a standard practice in mathematics, ensuring that the final answer is presented in its simplest form. It also makes it easier to compare and combine the result with other expressions.
This calculation highlights the importance of careful algebraic manipulation and attention to detail. Each step must be executed precisely to ensure that the final result is accurate. The use of the Pythagorean identity, combined with the quadrant information, allows us to unambiguously determine the value of sin(θ) from the given value of cos(θ).
Finally, Finding sin(2θ): Putting It All Together
With sin(θ) = √10/10 and cos(θ) = 3√10/10 in hand, we can now calculate sin(2θ) using the sine double-angle formula: sin(2θ) = 2sin(θ)cos(θ). Substituting the values we found, we get sin(2θ) = 2 * (√10/10) * (3√10/10). Multiplying the fractions, we have sin(2θ) = 2 * (3 * 10) / 100 = 60/100. Simplifying the fraction, we arrive at sin(2θ) = 3/5. This is the final answer, representing the sine of twice the angle θ.
The calculation of sin(2θ) demonstrates the power of combining trigonometric identities and algebraic techniques. By systematically applying the sine double-angle formula and the Pythagorean identity, we were able to solve a seemingly complex problem in a clear and concise manner. This approach highlights the importance of mastering fundamental trigonometric concepts and developing a strong problem-solving strategy.
The result, sin(2θ) = 3/5, provides a concrete example of how the double-angle formula can be used to relate trigonometric functions of different angles. This type of calculation is essential in many areas of mathematics and physics, including the analysis of oscillations, waves, and periodic phenomena. The ability to confidently apply these formulas is a valuable skill for anyone working in these fields.
Conclusion: Mastering Trigonometric Transformations
In conclusion, we have successfully calculated sin(2θ) given cos(θ) = 3√10/10 and the information that θ is in Quadrant I. This journey involved understanding the sine double-angle formula, leveraging the Pythagorean identity, and carefully considering the quadrant information to determine the correct sign of sin(θ). The final result, sin(2θ) = 3/5, exemplifies the power and elegance of trigonometric transformations. Mastering these techniques is crucial for anyone venturing into the realms of mathematics, physics, engineering, or any field where trigonometric functions play a pivotal role. The ability to manipulate trigonometric identities, solve trigonometric equations, and interpret trigonometric relationships is an invaluable asset for problem-solving and analytical thinking.
