Solving Trig Equations: Sum Of Solutions In Degrees

Hey guys! Today, we're diving into a trigonometric equation that might seem a bit intimidating at first glance, but trust me, we'll break it down step by step. Our goal is to find the sum of the solutions in degrees for the equation (1 + cos x) / sin x + sin x / (1 + cos x) = 4 within the interval [0°, 180°]. So, buckle up and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what we're dealing with. The equation involves trigonometric functions – cosine (cos x) and sine (sin x). We need to find the values of x that satisfy the equation, but only those values that fall within the range of 0° to 180°. Once we find those solutions, we'll add them up to get our final answer. Remember guys, the key to tackling any math problem is to break it down into manageable steps. This approach not only makes the problem less daunting but also helps in understanding the underlying concepts. Always start by identifying the core elements and the goal you're trying to achieve. In this case, the core elements are the trigonometric functions and the specified interval, and the goal is to find the sum of the solutions within that interval. By clearly defining these aspects, we set ourselves up for a more focused and efficient problem-solving process. Moreover, understanding the context of the problem—why we're solving it and what it represents—can provide additional insights and motivation. So, let's keep this structured approach in mind as we delve deeper into solving this trigonometric equation.

Step 1: Simplifying the Equation

The first thing we want to do is simplify the equation to make it easier to work with. We have two fractions added together, so let's find a common denominator. The common denominator for sin x and (1 + cos x) is simply their product: sin x * (1 + cos x). Now, let's rewrite the fractions with this common denominator:

(1 + cos x) / sin x + sin x / (1 + cos x) = [(1 + cos x)² + sin² x] / [sin x (1 + cos x)]

Now, let's expand the numerator:

[(1 + 2cos x + cos² x) + sin² x] / [sin x (1 + cos x)]

Remember the trigonometric identity: sin² x + cos² x = 1. We can use this to simplify the numerator further:

[1 + 2cos x + 1] / [sin x (1 + cos x)] = (2 + 2cos x) / [sin x (1 + cos x)]

We can factor out a 2 from the numerator:

2(1 + cos x) / [sin x (1 + cos x)]

Now, we can cancel out the (1 + cos x) terms, as long as cos x ≠ -1:

2 / sin x = 4

This simplification is crucial because it transforms a complex expression into a manageable form. Guys, mastering algebraic manipulations is essential for solving trigonometric equations. By simplifying, we reduce the chances of making errors and gain a clearer picture of the equation's structure. So, always look for opportunities to simplify – it's a game-changer!

Step 2: Solving for sin x

Now our equation is much simpler: 2 / sin x = 4. Let's solve for sin x:

sin x = 2 / 4

sin x = 1 / 2

Awesome! We've isolated sin x. This is a significant step because it allows us to focus on the values of x that satisfy this specific trigonometric condition. Guys, recognizing these fundamental steps is vital in problem-solving. Isolating the variable we're trying to find is a common strategy across various mathematical domains. In trigonometry, isolating trigonometric functions like sine, cosine, or tangent often simplifies the process of finding solutions. It's like narrowing down your search area – once you know what sin x equals, you can focus on the angles that produce that sine value. This approach not only makes the problem more manageable but also provides a clear path toward the solution. So, remember this technique: simplify the equation and isolate the trigonometric function to make the problem easier to solve.

Step 3: Finding the Solutions for x in the Interval [0°, 180°]

We need to find the angles x in the interval [0°, 180°] where sin x = 1/2. We know that sine corresponds to the y-coordinate on the unit circle. So, we're looking for angles where the y-coordinate is 1/2.

From our knowledge of the unit circle, we know that:

• sin 30° = 1/2
• sin 150° = 1/2

Both of these angles fall within our interval [0°, 180°]. Remember, sine is positive in both the first and second quadrants. Guys, visualizing the unit circle is super helpful when solving trigonometric equations. The unit circle gives us a clear picture of the values of sine, cosine, and tangent for different angles. Knowing the common angles and their corresponding trigonometric values is a must-have skill. When you see sin x = 1/2, you should immediately think of 30° and 150°. This intuition comes from practice and familiarity with the unit circle. If you're not quite there yet, don't worry! Keep practicing, and soon you'll be able to recall these values almost instantly. The unit circle is your friend in trigonometry – make sure you're well-acquainted!

Step 4: Checking for Extraneous Solutions

Remember when we canceled out (1 + cos x) in Step 1? We had the condition that cos x ≠ -1. Let's check if any of our solutions violate this condition.

• For x = 30°, cos 30° = √3 / 2 ≠ -1
• For x = 150°, cos 150° = -√3 / 2 ≠ -1

So, neither of our solutions violates the condition. Guys, checking for extraneous solutions is crucial! Whenever you cancel out terms or perform operations that could potentially introduce solutions that don't actually satisfy the original equation, you need to double-check. In this case, we had to make sure that our solutions didn't make the denominator zero in the original equation. It's a little extra work, but it can save you from getting the wrong answer. Think of it as being a detective – you're making sure all the pieces of the puzzle fit together perfectly. So, always be diligent and check for extraneous solutions!

Step 5: Calculating the Sum of the Solutions

Finally, we need to find the sum of our solutions:

Sum = 30° + 150° = 180°

Therefore, the sum of the solutions to the equation in the given interval is 180 degrees.

Conclusion

We did it! We successfully found the sum of the solutions to the trigonometric equation. Guys, remember that solving these types of problems involves a combination of algebraic manipulation, trigonometric identities, and a good understanding of the unit circle. Practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time! If you follow these steps carefully and practice regularly, you'll be able to tackle even the most challenging trigonometric problems with confidence. The key takeaways from this problem are the importance of simplifying equations, using trigonometric identities, visualizing the unit circle, checking for extraneous solutions, and, most importantly, practicing consistently. Each problem you solve adds to your understanding and skill set, so keep going, and you'll see improvement over time. And hey, if you ever get stuck, remember there are plenty of resources available, including textbooks, online tutorials, and study groups. So, don't hesitate to seek help when you need it. Math is a journey, and every step forward is a victory!