Exercise 39: Is -7π/4 A Valid Solution?

Hey guys! Let's dive into this question about Exercise 39. It looks like there's some confusion about whether -7π/4 is a valid answer, and it’s a great question to explore! Math can sometimes feel like navigating a maze, and angles are no exception. So, let’s break it down and make sure we understand the concepts involved. When we're dealing with angles, especially in trigonometry, we often encounter multiple ways to represent the same angle. This is because angles can be coterminal, meaning they share the same terminal side. Think of it like this: imagine you're rotating a line around a point. If you rotate it a certain amount, you reach a specific position. But if you keep rotating it, even going a full circle or more, you can end up in the same position again! This is the core idea behind coterminal angles.

To really grasp this, consider the unit circle. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. Angles are measured counterclockwise from the positive x-axis. One full rotation around the circle is 2π radians (or 360 degrees). So, if you rotate by an angle θ, and then rotate by another 2π, you end up at the same spot. This means θ and θ + 2π are coterminal angles. Similarly, you can rotate clockwise, which corresponds to negative angles. A clockwise rotation of 2π (or -360 degrees) also brings you back to the starting point. So, θ and θ - 2π are also coterminal. This concept is super important because it shows us that there isn't just one way to represent an angle. There are infinitely many coterminal angles for any given angle, and they all represent the same direction. Now, let's bring this back to the question about -7π/4. The key here is to understand how -7π/4 relates to other angles. A negative angle means we're rotating clockwise. -7π/4 represents a clockwise rotation of 7π/4 radians. But what does that look like on the unit circle? To figure this out, we can relate -7π/4 to a more familiar angle. We can add multiples of 2π to -7π/4 until we get an angle within the range of 0 to 2π (or 0 to 360 degrees), which is often easier to visualize. So, let's add 2π to -7π/4: -7π/4 + 2π = -7π/4 + 8π/4 = π/4. Ah ha! So -7π/4 is coterminal with π/4! This means that rotating clockwise by 7π/4 is the same as rotating counterclockwise by π/4. They both land you in the exact same spot on the unit circle. This is why understanding coterminal angles is so crucial. It allows us to see the relationships between different angle measures and to simplify problems by working with the most convenient representation of an angle. In this case, while -7π/4 might look a bit intimidating at first, recognizing that it's coterminal with π/4 makes it much easier to understand and work with.

Understanding Coterminal Angles

Let’s explore this concept of coterminal angles a little further. Think of it like finding different ways to say the same thing. In the world of angles, coterminal angles are different ways of describing the same direction. To find coterminal angles, you essentially add or subtract full rotations (multiples of 2π radians or 360 degrees) from a given angle. Why does this work? Well, adding or subtracting a full rotation just means you're going around the circle one or more times, ending up back where you started. So, the direction you're pointing in remains the same, even though the angle measure is different. Let's say you have an angle of θ. To find a coterminal angle, you can use the following formula: θ + 2πk, where k is any integer (positive, negative, or zero). If k is positive, you're adding rotations (going counterclockwise). If k is negative, you're subtracting rotations (going clockwise). If k is zero, you're not adding or subtracting any rotations, so you get the original angle back. For example, let's take an angle of π/3 radians. To find a coterminal angle, we can add 2π: π/3 + 2π = π/3 + 6π/3 = 7π/3. So, π/3 and 7π/3 are coterminal angles. They represent the same direction on the unit circle. We can also subtract 2π: π/3 - 2π = π/3 - 6π/3 = -5π/3. So, π/3 and -5π/3 are also coterminal angles. You can keep adding or subtracting 2π as many times as you want, and you'll always get a coterminal angle. This is why there are infinitely many coterminal angles for any given angle! Now, let's think about degrees. If you have an angle in degrees, you can find coterminal angles by adding or subtracting multiples of 360 degrees. So, the formula for coterminal angles in degrees is: θ + 360k, where k is any integer. For instance, let's take an angle of 60 degrees. To find a coterminal angle, we can add 360 degrees: 60 + 360 = 420. So, 60 degrees and 420 degrees are coterminal angles. We can also subtract 360 degrees: 60 - 360 = -300. So, 60 degrees and -300 degrees are also coterminal angles. Understanding coterminal angles is super useful for simplifying problems in trigonometry and calculus. Sometimes, working with a coterminal angle that's within a specific range (like 0 to 2π or 0 to 360 degrees) can make things much easier. It's like choosing the right tool for the job! In our original question about -7π/4, we used the concept of coterminal angles to simplify the angle and see its relationship to π/4. This is a common strategy in math, and it's a powerful one to have in your toolbox.

Applying the Concept to Exercise 39

Okay, let's bring it all back to Exercise 39! You asked specifically about question 2 and whether -7π/4 is a valid answer. Based on our discussion about coterminal angles, the answer is a resounding yes! -7π/4 is absolutely a valid answer. Remember, angles that are coterminal share the same terminal side, meaning they point in the same direction. -7π/4 and π/4 are coterminal, so they represent the same angle. The beauty of math is that there often isn't just one