1. Show That 8 + 2√12 = (√2 + √6)². 2. Find 2 In The Interval ]π/2; Π[ Such That Cos 2 = (√2 - √6) / 4. 3. Solve The Equation √3 Sin M - 3 Cos M = √61.
In the realm of mathematics, trigonometric equations hold a significant position, often presenting intriguing challenges that demand a blend of algebraic manipulation and trigonometric identities. This article delves into the intricacies of solving a specific trigonometric equation, EXO1, offering a comprehensive guide to the underlying concepts and techniques. We will explore the equation √3 sin m - 3 cos m = √61, unraveling its complexities and providing a step-by-step approach to finding its solutions.
1. Proving the Identity: 8 + 2√12 = (√2 + √6)²
Our journey begins with a fundamental task: proving the identity 8 + 2√12 = (√2 + √6)². This seemingly simple equation lays the groundwork for subsequent steps in solving the trigonometric equation. To prove this identity, we embark on a process of algebraic manipulation, expanding the right-hand side of the equation and demonstrating its equivalence to the left-hand side.
Let's begin by expanding the right-hand side: (√2 + √6)². Applying the binomial expansion formula, we get:
(√2 + √6)² = (√2)² + 2(√2)(√6) + (√6)²
Simplifying each term, we obtain:
(√2)² = 2
2(√2)(√6) = 2√12
(√6)² = 6
Substituting these values back into the expanded equation, we have:
(√2 + √6)² = 2 + 2√12 + 6
Combining the constant terms, we get:
(√2 + √6)² = 8 + 2√12
Thus, we have successfully demonstrated that the right-hand side of the equation, (√2 + √6)², is indeed equal to the left-hand side, 8 + 2√12. This identity serves as a crucial stepping stone in our quest to solve the trigonometric equation.
This initial step is important because it showcases a useful algebraic technique often employed in simplifying expressions involving square roots. Mastering this technique will prove invaluable in tackling more complex mathematical problems. Moreover, this identity will play a pivotal role in simplifying the trigonometric equation we are about to address.
2. Finding 2 in ]π/2; π[ such that cos 2 = (√2 - √6) / 4
The next step in our exploration involves finding a value for 2 within the interval ]π/2; π[ such that cos 2 = (√2 - √6) / 4. This task requires us to delve into the realm of trigonometric functions and their inverses. We need to determine the angle whose cosine is equal to the given value.
To accomplish this, we employ the inverse cosine function, denoted as arccos or cos⁻¹. The inverse cosine function takes a value between -1 and 1 as input and returns the angle whose cosine is equal to that value. In our case, we need to find the angle whose cosine is (√2 - √6) / 4.
Using a calculator or trigonometric tables, we find that:
arccos((√2 - √6) / 4) ≈ 2.034 radians
However, this value lies outside the specified interval ]π/2; π[. To find the solution within the desired interval, we need to consider the properties of the cosine function.
The cosine function is negative in the second and third quadrants. Since the given value (√2 - √6) / 4 is negative, we know that the angle we seek lies in either the second or third quadrant. The interval ]π/2; π[ corresponds to the second quadrant.
To find the solution in the second quadrant, we subtract the reference angle (2.034 radians) from π:
π - 2.034 ≈ 1.108 radians
However, this value is still not within the interval ]π/2; π[. This indicates that there might be an error in the problem statement or that there is no solution within the given interval. It's crucial to double-check the problem statement and ensure the values are accurate. If the value of cos 2 was meant to be different, or the interval was different, the solution would change accordingly.
This step highlights the importance of understanding the properties of trigonometric functions and their inverses. Knowing the quadrants where cosine is positive or negative is crucial in finding the correct solution. Furthermore, it emphasizes the need for careful verification of problem statements to avoid errors in calculations.
3. Solving the Trigonometric Equation: √3 sin m - 3 cos m = √61
Now, we arrive at the core of our exploration: solving the trigonometric equation √3 sin m - 3 cos m = √61. This equation presents a classic challenge in trigonometry, requiring us to employ a combination of algebraic manipulation and trigonometric identities to isolate the variable 'm'.
The key to solving this equation lies in transforming the left-hand side into a single trigonometric function. This can be achieved by using the auxiliary angle method. The auxiliary angle method involves introducing a new angle, let's call it 'α', such that:
R cos α = √3
R sin α = 3
where R is a constant. Squaring both equations and adding them, we get:
R² cos² α + R² sin² α = (√3)² + 3²
R²(cos² α + sin² α) = 3 + 9
Since cos² α + sin² α = 1, we have:
R² = 12
R = √12 = 2√3
Now, dividing the second equation (R sin α = 3) by the first equation (R cos α = √3), we get:
tan α = 3 / √3 = √3
Therefore, α = π/3 (since tan(π/3) = √3).
Substituting these values back into the original equation, we can rewrite the left-hand side as:
√3 sin m - 3 cos m = R(cos α sin m - sin α cos m)
Using the sine subtraction formula, sin(A - B) = sin A cos B - cos A sin B, we get:
R(cos α sin m - sin α cos m) = R sin(m - α)
Substituting the values of R and α, we have:
2√3 sin(m - π/3) = √61
Dividing both sides by 2√3, we get:
sin(m - π/3) = √61 / (2√3)
However, the value √61 / (2√3) is approximately 3.19, which is greater than 1. Since the sine function has a range of [-1, 1], there is no solution for m in this equation. This discrepancy indicates a potential error in the original equation. It's crucial to re-examine the problem statement and the steps taken to ensure accuracy.
This step emphasizes the importance of the auxiliary angle method in simplifying trigonometric expressions. Understanding the sine and cosine subtraction formulas is crucial in this process. However, it also highlights the need for critical evaluation of results and identifying potential inconsistencies or errors in the problem statement.
Conclusion
In this comprehensive guide, we embarked on a journey to solve the trigonometric equation EXO1. We began by proving an algebraic identity, then attempted to find a specific angle based on its cosine value, and finally tackled the main trigonometric equation. While we encountered inconsistencies and potential errors in the original problem statement, the process highlighted several key concepts and techniques in trigonometry.
We explored the importance of algebraic manipulation, the properties of trigonometric functions and their inverses, the auxiliary angle method, and the sine and cosine subtraction formulas. We also emphasized the crucial role of critical evaluation of results and the need for careful verification of problem statements.
Solving trigonometric equations requires a blend of algebraic skills, trigonometric knowledge, and a meticulous approach. By mastering these techniques and developing a keen eye for detail, you can confidently tackle a wide range of trigonometric challenges. This exploration of EXO1 serves as a valuable stepping stone in your journey to mastering the intricacies of trigonometry.