Finding Angles With Specific Properties: A Detailed Guide

Hey guys! Let's dive into a cool math problem that's all about angles. We're going to break down how to find out how many different angles fit specific criteria. This type of question often pops up in ÖSYM-style exams, so it's super important to understand the ins and outs. The question we're tackling is: "How many different 'a' angles are there, where 240° ≤ a ≤ 12000°, and the principal measure is 70°?" The answer choices are A) 31, B) 32, C) 33, D) 34, and E) 35. Sounds a bit tricky, right? Don't worry, we'll go through it step by step, making it easy to grasp. We'll use our knowledge of angles, their principal measures, and how they relate to each other. By the end, you'll be able to solve similar problems with confidence. So, let's get started!

Understanding the Basics: Angles and Their Principal Measures

Alright, before we jump into the main problem, let's refresh our memory on some fundamental concepts. Angles are formed when two rays meet at a common point, called the vertex. We measure angles in degrees (°). The principal measure of an angle is the angle's value when it's expressed within a specific range, usually between 0° and 360°. Think of it as the angle's simplest form within one full rotation. Any angle can have infinitely many values, but they all share the same principal measure. For example, 70°, 430°, and 790° all have a principal measure of 70°. This is because they all differ by multiples of 360°. So, if you keep adding or subtracting 360° from an angle, you'll get another angle with the same principal measure. Understanding this is key to solving the problem.

Principal Measure Explained

The principal measure is essentially the angle's equivalent within a single rotation (0° to 360°). To find the principal measure of any angle, we divide it by 360° and look at the remainder. The remainder is the principal measure. For example, to find the principal measure of 790°: 790° / 360° = 2 with a remainder of 70°. Therefore, the principal measure of 790° is 70°. Knowing how to find the principal measure is crucial for solving this type of problem. We're looking for angles within a specific range, but we're also given their principal measure, so we'll have to use this concept to work it out. Remembering that the principal measure is just the remainder after dividing by 360° will help you a lot!

Solving the Problem: Step-by-Step Guide

Now, let's get to the main part: solving the problem. The question wants us to find all the angles 'a' that meet two conditions: they fall within the range of 240° ≤ a ≤ 12000°, and their principal measure is 70°. Let's start with the principal measure condition. If the principal measure of an angle 'a' is 70°, it means that 'a' can be written as 70° plus a multiple of 360°. We can write this as a = 70° + 360°k, where 'k' is an integer. So, we're looking for different integer values of 'k' that fit the range of 'a' (240° to 12000°).

Setting Up the Inequality

Next, we'll incorporate the range condition (240° ≤ a ≤ 12000°) into our equation. Since a = 70° + 360°k, we can substitute this into the inequality: 240° ≤ 70° + 360°k ≤ 12000°. Now, our goal is to isolate 'k' to find the possible integer values that satisfy this inequality. To do this, we'll subtract 70° from all parts of the inequality: 240° - 70° ≤ 360°k ≤ 12000° - 70°. This simplifies to 170° ≤ 360°k ≤ 11930°.

Finding the Range of k

Now we're close! To get 'k' by itself, we'll divide all parts of the inequality by 360°. This gives us: 170° / 360° ≤ k ≤ 11930° / 360°. Let's calculate these values. 170° / 360° is approximately 0.47, and 11930° / 360° is approximately 33.14. So, we have 0.47 ≤ k ≤ 33.14. Remember, 'k' must be an integer. This means 'k' can be any whole number from 1 to 33, because the number must be greater than 0.47 and not exceed 33.14. To calculate the final answer, we need to determine the total number of integer values within the calculated range.

Counting the Solutions

Finally, let's figure out how many values 'k' can take. Since 'k' can be any integer from 1 to 33, we simply count the numbers in that range. The integers are 1, 2, 3, …, 33. To find the total number of integers, you can just look at the highest value, which is 33. Therefore, there are 33 possible values for 'k', and each value of 'k' corresponds to a different angle 'a' that satisfies the conditions. So, the correct answer is C) 33. This means that within the given range (240° to 12000°) and with a principal measure of 70°, there are 33 different angles.

Conclusion: Mastering Angle Problems

Awesome work, guys! We've successfully solved the problem. You've now seen how to find the number of angles based on their range and principal measure. Remember the key steps: understand the concept of principal measure, set up your equation (a = 70° + 360°k), incorporate the given range into your inequality, isolate 'k' to find its possible values, and then count the integer solutions. This approach works for all similar problems! Keep practicing, and you'll become a pro at these types of angle questions in no time. Good luck with your studies, and keep up the great work! You've got this!