Solving Complementary Angle Equations: A Trigonometry Guide

Hey guys! Today, we're diving into a fun problem involving complementary angles and some neat trigonometric identities. We're given a set of conditions and asked to find the value of an expression. Let's break down the problem step-by-step and see how we can solve it. Remember, understanding the core concepts is key! We will look at how to tackle complementary angles problems, providing a step-by-step solution. We'll start with the basics, define what complementary angles mean, and then move on to the more complex parts of the problem.

So, what are we waiting for? Let's jump right in!

Understanding Complementary Angles

Complementary angles, in simple terms, are two angles whose measures add up to 90 degrees (or π/2 radians). This is the foundation of our problem. This property is super important! If we have two angles, let's call them x and y, and they are complementary, then we can write this relationship as: x + y = 90°.

Knowing this opens the door to using some handy trigonometric identities. Remember that: sin(x) = cos(y), cos(x) = sin(y), tan(x) = cot(y), sec(x) = csc(y), and so on. These identities are going to be very helpful when solving the problem! For instance, if you know the value of sin(x), you immediately know the value of cos(y), as long as x and y are complementary.

Let’s explore this relationship further. In a right-angled triangle, the two acute angles are always complementary. This is because the sum of all angles in a triangle is 180°, and one angle is already 90°. So, the remaining two angles must sum up to 90°. This geometrical insight helps solidify your understanding. The beauty of complementary angles lies in their interconnectedness. Changing one angle directly affects the other, making them perfect partners in trigonometric relationships. The application of complementary angles extends beyond just the basics.

Consider real-world scenarios, such as the angles of elevation and depression in surveying or navigation. They are often complementary. Therefore, the principles we discuss today are not just academic but also practical. We'll encounter scenarios where we need to switch between sine and cosine, tangent and cotangent, etc. The key is to be familiar with the identities and to recognize when and how to apply them. It's like having a secret code to unlock the relationships between angles.

Setting Up the Problem

Alright, let’s get into the problem. We’re given that x and y are complementary angles, and we have the equation (tan x)^(cot² y) = 3√3. Our mission is to calculate E = 2sin(x/2) + sec(2y). The goal is to simplify this expression by using our knowledge of complementary angles and trigonometric identities. From the start, we know that x + y = 90°. Now, the equation gives us a relationship between tan x and cot y. Because x and y are complementary, we can rewrite cot y in terms of tan x. Remember that cot y = tan(90° - y) = tan x. This transformation simplifies the equation and allows us to get a numerical value, which is usually the key to solving the problem.

So, we will substitute y = 90° - x into the given equation (tan x)^(cot² y) = 3√3. This simplifies the equation significantly. Before we dive into the calculations, let's also think about the final expression we need to calculate: E = 2sin(x/2) + sec(2y). Here, we have sin(x/2) and sec(2y), which needs to be rewritten in terms of x. The equation tells us that x and y are complementary, meaning there's a direct relationship between them. Our first aim is to solve for x (or y). We can use this value to calculate E.

Solving for x and y

Let's work through this step-by-step to find the values of x and y first. We are given that (tan x)^(cot² y) = 3√3, and because x and y are complementary, x + y = 90°. Therefore, y = 90° - x. Now substitute y in the main equation: (tan x)^(cot² (90° - x)) = 3√3. Remember the important identity: cot(90° - x) = tan x. Replace the cotangent term: (tan x)^(tan² x) = 3√3.

Here's where things get interesting! Let’s think about what the base can be to equal to 3√3. We know that 3√3 can be written as 3^(3/2). Thus, we can deduce that tan x = √3, since (√3)^(√3²)= (√3)³ = 3√3. Since tan x = √3, x = arctan(√3) = 60° (or π/3 radians). We’ve found our value for x! Now that we have x, finding y is a breeze, because x + y = 90°. So, y = 90° - 60° = 30° (or π/6 radians). Now we have x = 60° and y = 30°. Keep an eye on what’s needed in the final equation. These are all the values we need to compute E, so we’re on the right track.

Calculating E

Now, let's calculate the value of E using the values of x and y we found. Our goal is to calculate: E = 2sin(x/2) + sec(2y). We've found that x = 60° and y = 30°. Let’s substitute these values into the expression for E. So, E = 2sin(60°/2) + sec(2*30°) = 2sin(30°) + sec(60°).

Now use your knowledge of trigonometric values: sin(30°) = 1/2 and sec(60°) = 2. So, E = 2*(1/2) + 2 = 1 + 2 = 3. We have now computed the answer! E = 3. Congratulations, guys, we made it! The expression E simplifies beautifully to a whole number. This often happens in mathematical problems; it’s a sign that we’re on the right track and that our work is correct. The final answer, E = 3, is the key takeaway of this problem.

Conclusion

Alright, folks, we've successfully navigated this problem involving complementary angles. We've seen how understanding the basic definition and identities related to complementary angles can make complex problems a lot easier. We started with the foundational concept of complementary angles and how to apply them. Then, we moved on to solving the equation to find x and y, using trigonometric identities to simplify the given equation. Finally, we substituted the values of x and y into the target expression, simplifying and getting the final result. Remember to review the key points: recognizing complementary angles, using related trigonometric identities, and making sure to carefully substitute the values.

Keep practicing! The more you work through these types of problems, the more comfortable you will become. Trigonometry can be a blast once you get the hang of it. If you have any questions, feel free to ask. Stay curious, keep learning, and keep practicing. Cheers and happy solving!