Calculate L = Sin²α - Cos²α Given Tan Α = Tan²(π/3)

In this article, we will delve into the process of calculating the value of L = sin²α - cos²α, given that α is an acute angle and tan α = tan²(π/3). This problem combines trigonometric identities and algebraic manipulation, offering a comprehensive exercise in mathematical problem-solving. We will break down each step, ensuring clarity and understanding for readers of all levels.

The primary keyword here is trigonometric identities, which form the bedrock of our solution. We'll also emphasize the concepts of acute angles, tangent function, sine function, and cosine function. The final calculation of L hinges on a clear understanding of these trigonometric principles and their interrelations. Our objective is not just to find the answer, but to provide a detailed, step-by-step guide that enhances your grasp of trigonometric problem-solving.

To kick things off, let's dissect the information we've been given. We know that α is an acute angle. What does that mean? Well, an acute angle is any angle that measures greater than 0 degrees but less than 90 degrees. In radians, this translates to 0 < α < π/2. This piece of information is crucial because it restricts α to the first quadrant, where all trigonometric functions (sine, cosine, tangent, etc.) are positive. Understanding this constraint is vital, as it influences how we interpret the values of trigonometric functions associated with α.

Next, we're told that tan α = tan²(π/3). This equation forms the core of our problem. To decipher this, we need to evaluate tan(π/3). Remember, π radians is equal to 180 degrees, so π/3 radians corresponds to 60 degrees. The tangent of 60 degrees, or tan(60°), is a standard trigonometric value that should be memorized or quickly derived using a 30-60-90 triangle. Tan(60°) is equal to √3. Thus, tan²(π/3) equals (√3)² = 3. So, our key equation simplifies to tan α = 3. This is a significant simplification, as it allows us to relate the tangent of α to a specific numerical value.

Now, let's think about what this means geometrically. The tangent of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Therefore, tan α = 3 tells us that the ratio of the opposite side to the adjacent side in a right-angled triangle with angle α is 3. This understanding paves the way for us to construct a right-angled triangle representation of α, which will be instrumental in determining the values of sin α and cos α.

The keyword here is acute angles, it constrains the range of α, making it lie in the first quadrant where trigonometric functions have positive values. The other keywords such as tangent function, sine function, and cosine function are also important, as they directly relate to the problem and the trigonometric relationships we will explore. By establishing these basic facts and trigonometric principles, we've set the stage for the next steps in solving the problem. The crucial understanding of acute angles and the values of trigonometric functions for standard angles is essential for further analysis. This foundational knowledge helps simplify the equation and provides a clear path towards finding the solution.

Given that tan α = 3, our next step is to find the values of sin α and cos α. To achieve this, we can employ a right-angled triangle approach. We know that tan α is the ratio of the opposite side to the adjacent side in a right-angled triangle. So, if we consider a right-angled triangle where the side opposite to angle α has a length of 3 units and the side adjacent to α has a length of 1 unit, then tan α will indeed be 3/1 = 3. This hypothetical triangle serves as a visual and conceptual aid in determining the sine and cosine of α.

Now, to find the lengths of all sides of the triangle, we need to calculate the hypotenuse. The Pythagorean theorem comes to our rescue here. The theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Applying this theorem to our triangle, we get:

hypotenuse² = opposite² + adjacent²
hypotenuse² = 3² + 1²
hypotenuse² = 9 + 1
hypotenuse² = 10
hypotenuse = √10

Thus, the length of the hypotenuse is √10 units. With the lengths of all three sides of the triangle now known, we can easily determine sin α and cos α. Recall that sin α is the ratio of the opposite side to the hypotenuse, and cos α is the ratio of the adjacent side to the hypotenuse. Therefore:

sin α = opposite / hypotenuse = 3 / √10
cos α = adjacent / hypotenuse = 1 / √10

These values of sin α and cos α are crucial for the final calculation. Note that both sin α and cos α are positive, which aligns with the fact that α is an acute angle (0 < α < π/2), and all trigonometric functions are positive in the first quadrant. The keyword right-angled triangle is central to our approach here, as it provides a visual and geometric context for understanding the ratios of trigonometric functions. The Pythagorean theorem is another crucial concept, allowing us to calculate the length of the hypotenuse, which is essential for determining sine and cosine. Also, the sine function and cosine function definitions are used to calculate the sine and cosine values which are key to solving the problem. Using trigonometric relationships and the geometry of right triangles, we've successfully found the values of sin α and cos α. This sets the stage for the final calculation of L.

Now that we've determined the values of sin α and cos α, we can proceed to calculate the value of L, which is defined as L = sin²α - cos²α. We found that sin α = 3 / √10 and cos α = 1 / √10. To find sin²α and cos²α, we simply square these values:

sin²α = (3 / √10)² = 9 / 10
cos²α = (1 / √10)² = 1 / 10

Now, we substitute these values into the expression for L:

L = sin²α - cos²α
L = (9 / 10) - (1 / 10)
L = 8 / 10
L = 0.8

Therefore, the value of L is 0.8. This completes our calculation, providing the solution to the problem. It's important to note the sequence of steps we followed to arrive at this answer. We began by understanding the given information, which included the fact that α is an acute angle and the equation tan α = tan²(π/3). We then simplified this equation to tan α = 3. Using this information, we constructed a right-angled triangle to find the values of sin α and cos α. Finally, we substituted these values into the expression L = sin²α - cos²α to obtain the result. The keyword here, L = sin²α - cos²α, defines the quantity we aim to calculate. The key concepts used here are sine function and cosine function, which gives us the final answer 0.8. This final step demonstrates how a combination of trigonometric understanding and algebraic manipulation can lead to the solution of a complex problem.

In summary, we have successfully calculated the value of L = sin²α - cos²α given that α is an acute angle and tan α = tan²(π/3). The solution involved a series of steps, each building upon the previous one. We started by recognizing that tan²(π/3) equals 3, which simplified the given condition to tan α = 3. Then, we utilized a right-angled triangle to visualize the relationship between the sides and the angle α, allowing us to determine sin α and cos α. The Pythagorean theorem was instrumental in finding the hypotenuse of this triangle. Finally, we substituted the values of sin α and cos α into the expression for L, arriving at the answer L = 0.8.

This problem exemplifies how a solid grasp of fundamental trigonometric concepts and identities, combined with algebraic skills, can lead to the resolution of complex mathematical challenges. The acute angle condition was crucial in restricting the possible values of trigonometric functions, and the understanding of tangent, sine, and cosine in the context of a right-angled triangle was key to finding the solution. The keyword conclusion emphasizes the summary of the entire problem-solving process, reinforcing the key steps and concepts involved. Overall, this exercise demonstrates the interconnectedness of various mathematical concepts and the power of systematic problem-solving strategies.

Final Answer: e) 0.8