Solving Trigonometric Equations: Unveiling The (a, B) Pair

Hey math enthusiasts! Today, we're diving into a fascinating trigonometric problem that's sure to tickle your problem-solving senses. We're given a rather intriguing equation involving trigonometric functions and tasked with finding the ordered pair (a, b). Let's break down the problem step by step, unraveling the intricacies of trigonometric identities and algebraic manipulations.

Unpacking the Trigonometric Puzzle

Our journey begins with the given equation:

sin x + cos x + tan x + cot x + sec x + csc x = 7

And we are also provided with sin 2x = a - b√7. Our ultimate goal is to figure out the values of a and b that satisfy the given conditions. This kind of problem is a classic example of how different areas of mathematics, in this case, trigonometry and algebra, often intertwine. It requires a good understanding of trigonometric identities, algebraic manipulation techniques, and the ability to connect the dots to arrive at a solution. The beauty of these problems lies in the elegance of the solution, which often involves simplifying complex expressions into more manageable forms.

Let's take a closer look at the given equation. We have a sum of six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. The presence of these functions suggests that we'll likely need to utilize various trigonometric identities to simplify the equation. Trigonometric identities are essentially equations that hold true for all values of the variables involved. They are the building blocks for solving many trigonometry problems. Remember the basic definitions: tan x = sin x / cos x, cot x = cos x / sin x, sec x = 1 / cos x, and csc x = 1 / sin x. These definitions are fundamental to manipulating the given equation. Our strategy is to rewrite the equation in terms of sine and cosine, as this will help us in simplifying and eventually solving it.

To effectively solve this equation, it's crucial to have a strong grasp of trigonometric identities. For instance, the Pythagorean identity (sin²x + cos²x = 1) is a cornerstone of trigonometric manipulations. Then, there are the angle addition and subtraction formulas (e.g., sin(x + y) = sin x cos y + cos x sin y), which can be helpful in simplifying expressions involving sums and differences of angles. Furthermore, double-angle formulas (e.g., sin 2x = 2 sin x cos x) are essential for simplifying expressions involving multiples of angles. The more familiar you are with these identities, the more easily you will be able to navigate the intricacies of the problem. This problem is not just about finding a solution; it's about appreciating the underlying mathematical relationships and the power of simplification.

Transforming and Simplifying the Equation

Let's start by rewriting the equation, keeping in mind our ultimate aim: to express everything in terms of sine and cosine.

sin x + cos x + (sin x / cos x) + (cos x / sin x) + (1 / cos x) + (1 / sin x) = 7

Next, let's group the terms to make the simplification process more organized. Combining the fractions, we get:

sin x + cos x + (sin²x + cos²x) / (sin x cos x) + (sin x + cos x) / (sin x cos x) = 7

Notice that sin²x + cos²x = 1. This is where the Pythagorean identity comes in handy! Substituting this into our equation, we obtain:

sin x + cos x + 1 / (sin x cos x) + (sin x + cos x) / (sin x cos x) = 7

Now, let's make a substitution to simplify things. Let u = sin x + cos x and also note that sin 2x = 2 sin x cos x. We can rewrite the equation as:

u + 1 / (sin x cos x) + u / (sin x cos x) = 7

We know sin 2x = 2 sin x cos x, so sin x cos x = sin 2x / 2. Substituting this in:

u + 1 / (sin 2x / 2) + u / (sin 2x / 2) = 7

Simplifying this gives us:

u + 2 / sin 2x + 2u / sin 2x = 7

Further, u = sin x + cos x. Squaring both sides, we get u² = sin²x + 2 sin x cos x + cos²x. This simplifies to u² = 1 + sin 2x, or sin 2x = u² - 1. Substituting this into our equation, it becomes:

u + 2 / (u² - 1) + 2u / (u² - 1) = 7

This is a significant simplification. Now we have a single equation in terms of u.

Solving for u and sin 2x

To solve for u, let's rearrange the terms and solve the quadratic equation.

u + (2 + 2u) / (u² - 1) = 7

Multiplying both sides by (u² - 1):

u(u² - 1) + 2 + 2u = 7(u² - 1)

Expanding and rearranging:

u³ - u + 2 + 2u = 7u² - 7

u³ - 7u² + u + 9 = 0

By observation or using the rational root theorem, we find that u = -1 is a root. So, (u + 1) is a factor. Dividing the cubic equation by (u + 1) gives us:

(u + 1)(u² - 8u + 9) = 0

Therefore, we have u = -1 or u² - 8u + 9 = 0.

Solving the quadratic equation using the quadratic formula, u = (8 ± √(64 - 36)) / 2, which simplifies to u = 4 ± √7.

We know that u = sin x + cos x, and the range of sin x + cos x is [-√2, √2]. Therefore, the only possible value for u is u = 4 - √7. Hence, u = sin x + cos x = 4 - √7.

We have previously established that sin 2x = u² - 1. Substituting the value of u:

sin 2x = (4 - √7)² - 1

sin 2x = (16 - 8√7 + 7) - 1

sin 2x = 23 - 8√7

Thus, comparing with sin 2x = a - b√7, we can identify that a = 23 and b = 8. This gives us our ordered pair.

Unveiling the Final Answer

Therefore, the ordered pair (a, b) that satisfies the given conditions is (23, 8).

This problem beautifully demonstrates the power of trigonometric identities, algebraic manipulation, and the importance of understanding the ranges of trigonometric functions. It also emphasizes the importance of making strategic substitutions to simplify complex equations. Remember, with practice and a solid understanding of fundamental concepts, even the most challenging trigonometric problems can be solved. Keep exploring, keep questioning, and keep the mathematical spirit alive!

This journey through the trigonometric equation has shown how each step, from simplification to solving, is crucial. The strategic use of trigonometric identities, the smart substitutions, and the careful algebraic manipulations all play a vital role. In the end, it’s not just about the answer (23, 8); it's about the entire process, the problem-solving skills you gain, and the deeper understanding of mathematics that you achieve. Every problem you solve enriches your mathematical toolkit and enhances your ability to tackle future challenges. Remember to always look for patterns, use known identities, and simplify expressions to their most manageable forms.

Let's continue to delve into the fascinating world of mathematics, exploring its depths, and appreciating its elegance. Until our next mathematical adventure!

For further exploration, you can visit Khan Academy's Trigonometry Section.