Finding Cos(2θ) Given Cos(θ) = -5/13 And Θ In Quadrant II
In this comprehensive article, we will delve into the process of finding the value of cos(2θ) given that cos(θ) = -5/13 and θ lies in Quadrant II. This is a classic trigonometry problem that requires a solid understanding of trigonometric identities and the properties of angles in different quadrants. We will break down the problem step by step, providing clear explanations and justifications for each step. By the end of this article, you will have a firm grasp of how to solve such problems and a deeper appreciation for the elegance of trigonometry.
Understanding the Problem
Before we dive into the calculations, let's take a moment to understand the problem statement. We are given that cos(θ) = -5/13. This tells us the ratio of the adjacent side to the hypotenuse in a right triangle formed by the angle θ. The negative sign indicates that the angle θ lies in either Quadrant II or Quadrant III, where the x-coordinate (and hence the cosine) is negative. We are further given that θ is in Quadrant II. This crucial piece of information narrows down the possibilities and allows us to determine the sign of other trigonometric functions of θ, such as sin(θ), which will be positive in Quadrant II.
Our goal is to find cos(2θ), which is the cosine of twice the angle θ. To do this, we will utilize the double angle formula for cosine. This formula expresses cos(2θ) in terms of cos(θ) and sin(θ). Since we already know cos(θ), we need to find sin(θ) to apply the formula. We will use the Pythagorean identity to find sin(θ), and then we can plug the values of cos(θ) and sin(θ) into the double angle formula to get our final answer.
Understanding the context of the problem is paramount. Trigonometric functions are not just abstract mathematical concepts; they have real-world applications in fields like physics, engineering, and navigation. Angles in standard position, where the initial side lies along the positive x-axis, are fundamental in describing rotations and oscillations. The quadrant in which an angle lies dictates the signs of its trigonometric functions, and this is essential for accurate calculations and interpretations. By understanding the underlying principles, we can approach these problems with confidence and clarity.
Utilizing the Pythagorean Identity
To find sin(θ), we will employ the fundamental Pythagorean identity, which states that sin²(θ) + cos²(θ) = 1. This identity is a cornerstone of trigonometry and is derived directly from the Pythagorean theorem applied to the unit circle. It connects the sine and cosine functions in a fundamental way, allowing us to find one if we know the other.
We are given that cos(θ) = -5/13. Substituting this value into the Pythagorean identity, we get:
sin²(θ) + (-5/13)² = 1
Simplifying, we have:
sin²(θ) + 25/169 = 1
Subtracting 25/169 from both sides, we get:
sin²(θ) = 1 - 25/169
sin²(θ) = 144/169
Now, we take the square root of both sides:
sin(θ) = ±√(144/169)
sin(θ) = ±12/13
We have two possible values for sin(θ): +12/13 and -12/13. However, we know that θ is in Quadrant II, where the sine function is positive. Therefore, we choose the positive value:
sin(θ) = 12/13
The Pythagorean identity is a powerful tool, but it's crucial to remember that taking the square root introduces two possible solutions, a positive and a negative one. We must use the information about the quadrant to determine the correct sign. In this case, knowing that θ is in Quadrant II allowed us to confidently choose the positive value for sin(θ). This highlights the importance of understanding the properties of trigonometric functions in different quadrants.
Applying the Double Angle Formula
Now that we have found sin(θ) = 12/13, we can proceed to calculate cos(2θ) using the double angle formula for cosine. There are three common forms of the double angle formula for cosine:
- cos(2θ) = cos²(θ) - sin²(θ)
- cos(2θ) = 2cos²(θ) - 1
- cos(2θ) = 1 - 2sin²(θ)
Any of these formulas will give us the correct answer. Let's use the first formula, cos(2θ) = cos²(θ) - sin²(θ), since we already have the values of cos(θ) and sin(θ).
Substituting cos(θ) = -5/13 and sin(θ) = 12/13 into the formula, we get:
cos(2θ) = (-5/13)² - (12/13)²
cos(2θ) = 25/169 - 144/169
cos(2θ) = -119/169
Therefore, cos(2θ) = -119/169. This is our final answer. We could have also used the other two forms of the double angle formula, and we would have arrived at the same result. For example, using cos(2θ) = 2cos²(θ) - 1, we get:
cos(2θ) = 2(-5/13)² - 1
cos(2θ) = 2(25/169) - 1
cos(2θ) = 50/169 - 1
cos(2θ) = 50/169 - 169/169
cos(2θ) = -119/169
As expected, we get the same answer. The double angle formulas are powerful tools for simplifying trigonometric expressions and solving equations. Understanding these formulas and knowing when to apply them is crucial for success in trigonometry.
Conclusion
In this article, we have successfully found the value of cos(2θ) given that cos(θ) = -5/13 and θ is in Quadrant II. We accomplished this by first using the Pythagorean identity to find sin(θ), carefully considering the quadrant in which θ lies to determine the correct sign. Then, we applied the double angle formula for cosine to calculate cos(2θ). The final answer is cos(2θ) = -119/169.
This problem highlights the importance of understanding trigonometric identities, the properties of angles in different quadrants, and the relationships between trigonometric functions. By mastering these concepts, you can confidently tackle a wide range of trigonometry problems. Trigonometry is a fundamental branch of mathematics with numerous applications, and a solid understanding of its principles is essential for success in many scientific and technical fields. Practice and careful attention to detail are key to mastering trigonometry and its applications. Remember to always consider the quadrant when determining the signs of trigonometric functions, and don't hesitate to use the various forms of trigonometric identities to simplify your calculations.
