Let's break down this trigonometric problem step by step. We aim to simplify and solve the expression: cot(40°) * tan(50°) + cos(35°) / (2sin(55°)). This involves understanding trigonometric identities and relationships between different angles. So, grab your thinking caps, guys, and let’s dive in!

Understanding the Trigonometric Functions

Before we start crunching numbers, let's quickly recap what each trigonometric function represents:

• Cotangent (cot): This is the reciprocal of the tangent function. So, cot(x) = 1 / tan(x) = cos(x) / sin(x).
• Tangent (tan): This is the ratio of the sine to cosine. So, tan(x) = sin(x) / cos(x).
• Cosine (cos): In a right triangle, this is the ratio of the adjacent side to the hypotenuse.
• Sine (sin): In a right triangle, this is the ratio of the opposite side to the hypotenuse.

These functions are the building blocks of our problem, and knowing their definitions helps in simplifying complex expressions. We will use these definitions and some trigonometric identities to simplify the given expression and arrive at a solution. Make sure you have a strong grasp of these basics, as they are fundamental to solving trigonometric problems. The relationships between these functions often allow us to rewrite expressions in a more manageable form, which is key to solving the problem at hand. Remembering these identities will make the whole process smoother and less intimidating. With a solid understanding of these trigonometric functions, we can confidently tackle the problem.

Simplifying cot(40°) * tan(50°)

The first part of our expression is cot(40°) * tan(50°). To simplify this, we need to recognize a key relationship between cotangent and tangent functions, as well as how angles relate to each other. Remember that cot(x) = 1 / tan(x). Also, notice that 40° and 50° add up to 90°, meaning they are complementary angles. Here’s how we can simplify this part:

Since 40° + 50° = 90°, we can say that 50° = 90° - 40°. Therefore, tan(50°) = tan(90° - 40°). Using the identity tan(90° - x) = cot(x), we get tan(50°) = cot(40°). Now we can rewrite the first part of the expression:

cot(40°) * tan(50°) = cot(40°) * cot(40°). Whoops! It should be: cot(40°) * tan(50°) = cot(40°) * tan(90°-40°) = cot(40°) * cot(40°)

So: cot(40°) * tan(50°) = cot(40°) * cot(40°) = cot(40°) * (1 / tan(40°))

Because cot(40°) is the reciprocal of tan(40°), cot(40°) * tan(40°) equals 1. So: cot(40°) * tan(50°) = 1. This simplification is crucial because it significantly reduces the complexity of the original expression. Recognizing the complementary angle relationship and applying the appropriate trigonometric identity allowed us to simplify the expression to a simple value. This showcases the power of using trigonometric identities to make problems more manageable. Always be on the lookout for such relationships, as they can greatly simplify your calculations and lead you to the solution more efficiently. Thus, simplifying cot(40°) * tan(50°) to 1 is a key step in solving the overall problem.

Simplifying cos(35°) / (2sin(55°))

Now let's tackle the second part of our expression: cos(35°) / (2sin(55°)). Again, we'll use the concept of complementary angles to simplify this. Notice that 35° + 55° = 90°, which means 35° and 55° are complementary angles. We can use the identity sin(x) = cos(90° - x). Therefore, sin(55°) = cos(90° - 55°) = cos(35°).

Now we can rewrite the second part of the expression:

cos(35°) / (2sin(55°)) = cos(35°) / (2cos(35°)).

Here, cos(35°) appears in both the numerator and the denominator, so we can cancel it out:

cos(35°) / (2cos(35°)) = 1 / 2.

This simplification is incredibly useful because it turns a complex trigonometric ratio into a simple fraction. By recognizing the complementary angle relationship and applying the appropriate identity, we've made this part of the problem much easier to handle. It’s a great illustration of how identifying and using trigonometric relationships can significantly simplify complex expressions. Simplifying cos(35°) / (2sin(55°)) to 1/2 is another crucial step in solving the overall problem. This step highlights the importance of recognizing complementary angles and utilizing trigonometric identities to simplify expressions effectively. Remember, always look for ways to simplify trigonometric expressions using these techniques to make problem-solving easier.

Combining the Simplified Terms

We've simplified both parts of the original expression. Now let’s combine them:

Original expression: cot(40°) * tan(50°) + cos(35°) / (2sin(55°))

Simplified first part: cot(40°) * tan(50°) = 1

Simplified second part: cos(35°) / (2sin(55°)) = 1 / 2

So, the combined expression is:

1 + 1 / 2 = 3 / 2

Therefore, cot(40°) * tan(50°) + cos(35°) / (2sin(55°)) = 3 / 2. This is the final simplified answer. Combining the simplified terms is a straightforward step, but it’s important to ensure that each part has been simplified correctly before adding them together. By breaking down the problem into smaller, manageable parts and then combining the results, we've successfully solved the original expression. This approach highlights the importance of methodical problem-solving and attention to detail. Double-checking each step ensures accuracy and confidence in the final answer. Always remember to combine the simplified terms carefully to arrive at the correct solution.

Final Answer

So, after simplifying each component and combining them, we find that:

cot(40°) * tan(50°) + cos(35°) / (2sin(55°)) = 3 / 2.

In conclusion, solving complex trigonometric expressions often involves breaking them down into smaller parts, applying trigonometric identities, and simplifying each part before combining them. Understanding the relationships between trigonometric functions and recognizing complementary angles are key skills in this process. Always remember to double-check your work and ensure that each step is accurate. This systematic approach will help you tackle even the most challenging trigonometric problems with confidence.

Therefore, the final answer is 3/2.