Finding The Value Of Sec Θ Given Tan² Θ = 3/8
In trigonometry, understanding the relationships between different trigonometric functions is crucial for solving various problems. This article delves into a specific problem involving the tangent and secant functions. Our main task is to find the value of sec θ given that tan² θ = 3/8. This involves leveraging trigonometric identities and algebraic manipulation. This detailed explanation aims to provide a clear understanding of the steps involved, making it accessible and informative for students and enthusiasts alike. We will explore the fundamental trigonometric identity linking secant and tangent, discuss the importance of considering both positive and negative roots, and meticulously work through the calculations to arrive at the correct solution. This article not only provides the answer but also reinforces the underlying concepts, ensuring a solid grasp of trigonometric principles. Furthermore, this exploration underscores the significance of careful algebraic manipulation and the consideration of all possible solutions in trigonometric problems. Understanding these concepts is essential for more advanced topics in mathematics and physics, where trigonometric functions play a vital role. Therefore, let's embark on this journey to unravel the solution step by step, ensuring a comprehensive understanding of the problem and its resolution.
Understanding the Trigonometric Identity
To solve the problem, we need to utilize a fundamental trigonometric identity that connects the tangent and secant functions. This identity is derived from the Pythagorean identity, which is a cornerstone of trigonometry. The Pythagorean identity states that sin² θ + cos² θ = 1. By dividing this entire equation by cos² θ, we can derive the relationship between tangent and secant. Recall that tan θ = sin θ / cos θ and sec θ = 1 / cos θ. Therefore, when we divide the Pythagorean identity by cos² θ, we get:
(sin² θ / cos² θ) + (cos² θ / cos² θ) = 1 / cos² θ
This simplifies to:
tan² θ + 1 = sec² θ
This identity is crucial because it directly links the square of the tangent function to the square of the secant function. Knowing this identity allows us to easily switch between tangent and secant values, which is exactly what we need to solve our problem. This relationship is not just a formula to memorize; it is a fundamental concept that arises directly from the geometry of the unit circle and the definitions of trigonometric functions. Understanding how this identity is derived from the basic Pythagorean identity provides a deeper comprehension of trigonometric relationships. Moreover, this identity is used extensively in various branches of mathematics and physics, particularly in calculus and mechanics, where trigonometric functions are used to model periodic phenomena. Therefore, mastering this identity is not just about solving this particular problem but about building a strong foundation for further studies in related fields. In the next sections, we will apply this identity to the given problem and carefully work through the steps to find the value of sec θ.
Applying the Identity to Find sec² θ
Now that we have the trigonometric identity tan² θ + 1 = sec² θ, we can substitute the given value of tan² θ into the equation. The problem states that tan² θ = 3/8. Plugging this value into the identity, we get:
(3/8) + 1 = sec² θ
To simplify this, we need to add the fraction 3/8 to the whole number 1. We can rewrite 1 as 8/8, so the equation becomes:
(3/8) + (8/8) = sec² θ
Adding the fractions, we get:
(3 + 8) / 8 = sec² θ
This simplifies to:
11/8 = sec² θ
So, we have found that sec² θ = 11/8. This is a crucial intermediate step, but we are not done yet. We need to find the value of sec θ, not sec² θ. To do this, we need to take the square root of both sides of the equation. Taking the square root is a fundamental algebraic operation, but it is important to remember that it can yield both positive and negative solutions. This is particularly important in trigonometric problems, where the sign of the function depends on the quadrant in which the angle θ lies. In the next section, we will carefully consider both the positive and negative square roots to find the possible values of sec θ.
Finding sec θ by Taking the Square Root
We have established that sec² θ = 11/8. To find sec θ, we need to take the square root of both sides of this equation. Remember that when we take the square root, we must consider both the positive and negative roots. This is because both the positive and negative values, when squared, will give the same positive result. Therefore, we have:
sec θ = ±√(11/8)
This means that sec θ can be either the positive square root of 11/8 or the negative square root of 11/8. This is a crucial point in solving trigonometric problems, as the sign of the trigonometric function depends on the quadrant in which the angle θ lies. The secant function is positive in the first and fourth quadrants and negative in the second and third quadrants. Therefore, without additional information about the angle θ, we must consider both possibilities. Now, let's simplify the square root. We have:
sec θ = ±√(11) / √(8)
We can simplify √(8) further by recognizing that 8 is 4 times 2, and 4 is a perfect square. So, √(8) = √(4 * 2) = √(4) * √(2) = 2√(2). Thus, we have:
sec θ = ±√(11) / (2√(2))
To rationalize the denominator, we multiply both the numerator and the denominator by √(2):
sec θ = ±(√(11) * √(2)) / (2√(2) * √(2))
This simplifies to:
sec θ = ±√(22) / (2 * 2)
sec θ = ±√(22) / 4
Thus, the possible values for sec θ are √(22) / 4 and -√(22) / 4. However, looking at the answer choices provided, we see that the solution is expressed as ±√(11/8). This is the unsimplified form we found earlier, which is perfectly correct. Therefore, the final answer is ±√(11/8). This careful consideration of both positive and negative roots and simplification of the square root ensures that we arrive at the correct solution.
Conclusion
In summary, given that tan² θ = 3/8, we found the value of sec θ by using the trigonometric identity tan² θ + 1 = sec² θ. We substituted the given value of tan² θ into the identity, which gave us sec² θ = 11/8. Taking the square root of both sides, we obtained sec θ = ±√(11/8). This highlights the importance of considering both positive and negative roots when solving trigonometric equations. The final answer, ±√(11/8), reflects the two possible values for sec θ depending on the quadrant in which the angle θ lies. This exercise underscores the significance of understanding fundamental trigonometric identities and applying algebraic manipulations correctly. Furthermore, it emphasizes the need to consider all possible solutions when dealing with square roots in trigonometric contexts. By carefully working through each step, from substituting the given value to simplifying the result, we have demonstrated a comprehensive approach to solving trigonometric problems. This method not only provides the correct answer but also reinforces the underlying principles of trigonometry, enabling a deeper understanding of these essential concepts. The ability to apply these principles is crucial for success in more advanced mathematical studies and in various applications of trigonometry in physics and engineering. Therefore, mastering these fundamental techniques is an invaluable asset for students and professionals alike.
Answer
The correct answer is B. ±√(11/8).
