Reference Angle For 240 Degrees A Step By Step Solution

Hey everyone! Today, we're diving into a fundamental concept in trigonometry: reference angles. Specifically, we're going to figure out the reference angle for a 240° angle. This is a super important skill for anyone studying trigonometry, as reference angles help simplify calculations and make understanding trigonometric functions much easier. So, let's get started and break it down step by step!

Understanding Reference Angles

Before we jump into solving for the reference angle of 240°, let's first understand what a reference angle actually is. In simple terms, a reference angle is the acute angle (an angle less than 90°) formed between the terminal side of a given angle and the x-axis. Think of it as the shortest distance from the terminal side of your angle to the x-axis. This concept helps us relate angles in different quadrants to angles in the first quadrant, which makes trigonometric calculations way more manageable.

Why are reference angles so important, you ask? Well, trigonometric functions like sine, cosine, and tangent have the same absolute values for angles with the same reference angle. The only thing that changes is the sign (positive or negative), which depends on the quadrant. So, by finding the reference angle, we can simplify the process of finding trigonometric values for any angle.

To really nail this concept, let's walk through how to find reference angles in each of the four quadrants:

• Quadrant I (0° - 90°): If your angle falls in the first quadrant, the reference angle is simply the angle itself. Easy peasy!
• Quadrant II (90° - 180°): In the second quadrant, you subtract the given angle from 180° to find the reference angle. The formula here is: Reference Angle = 180° - Given Angle.
• Quadrant III (180° - 270°): When your angle is in the third quadrant, you subtract 180° from the given angle. The formula is: Reference Angle = Given Angle - 180°.
• Quadrant IV (270° - 360°): Finally, in the fourth quadrant, you subtract the given angle from 360° to find the reference angle. The formula is: Reference Angle = 360° - Given Angle.

Knowing these rules is half the battle. Now, let's apply this knowledge to our specific problem: finding the reference angle for 240°.

Determining the Quadrant of 240°

The first step in finding the reference angle is to determine which quadrant the angle lies in. This is crucial because the formula we use to calculate the reference angle depends on the quadrant.

Remember, the quadrants are divided as follows:

• Quadrant I: 0° to 90°
• Quadrant II: 90° to 180°
• Quadrant III: 180° to 270°
• Quadrant IV: 270° to 360°

Looking at these ranges, it's clear that 240° falls between 180° and 270°, placing it firmly in the third quadrant. This is a key piece of information because it tells us which formula to use for finding the reference angle.

Visualizing this on the unit circle can be incredibly helpful. Imagine starting at the positive x-axis (0°) and rotating counterclockwise. By the time you reach 240°, you've gone past 180° but haven't quite made it to 270°, confirming that we're indeed in Quadrant III. Getting comfortable with this mental picture will make identifying quadrants a breeze.

Now that we've pinpointed the quadrant, we're ready to move on to the next step: applying the correct formula to calculate the reference angle. Knowing the quadrant is like having the right map – it guides us to the correct solution. So, let's use our map and find that reference angle!

Calculating the Reference Angle for 240°

Alright, guys, we've established that 240° lies in the third quadrant. Now comes the fun part: calculating the reference angle! As we discussed earlier, the formula for finding the reference angle in the third quadrant is:

Reference Angle = Given Angle - 180°

This formula essentially measures how far past 180° our angle goes. In our case, the given angle is 240°, so we simply plug that into the formula:

Reference Angle = 240° - 180°

Now, let's do the math. 240 minus 180 equals 60. So:

Reference Angle = 60°

There you have it! The reference angle for 240° is 60°.

This means that the acute angle formed between the terminal side of the 240° angle and the x-axis is 60°. Remember, reference angles are always acute angles, meaning they are between 0° and 90°. Our result of 60° fits perfectly within this range, which gives us confidence that we've calculated it correctly.

To double-check our work, we can visualize this on the unit circle. The 240° angle is 60° past the 180° mark, confirming our calculation. This visual check is a great way to reinforce your understanding and ensure accuracy.

So, we've not only found the reference angle but also understood why and how it works. Let's recap what we've learned and then look at the answer choices.

Reviewing the Answer Choices

Okay, we've successfully calculated that the reference angle for 240° is 60°. Now, let's take a look at the answer choices provided:

A. 30° B. 60° C. 180° D. 270°

It's pretty clear which one is the correct answer, isn't it? Option B, 60°, matches our calculation perfectly. This confirms that we've gone through the steps correctly and arrived at the right solution.

Let's quickly analyze why the other options are incorrect:

• A. 30°: This is not the correct reference angle for 240°. It's a common mistake to confuse reference angles with other angles, but we've clearly shown that 60° is the correct answer.
• C. 180° and D. 270°: These are not acute angles (less than 90°) and therefore cannot be reference angles. Remember, reference angles are always the acute angle formed with the x-axis.

By understanding why the incorrect options are wrong, we solidify our understanding of reference angles even further. This kind of analysis helps prevent similar mistakes in the future.

So, with confidence, we can select option B as the correct answer. We've not only found the solution but also gained a deeper understanding of the concept along the way.

Final Answer

Alright, team! We've journeyed through the world of reference angles and successfully found the reference angle for 240°. To recap, we:

1. Defined what reference angles are and why they're important.
2. Learned how to find reference angles in each of the four quadrants.
3. Determined that 240° lies in the third quadrant.
4. Calculated the reference angle using the formula for Quadrant III: Reference Angle = Given Angle - 180°.
5. Arrived at the solution: Reference Angle = 60°.
6. Reviewed the answer choices and confidently selected the correct answer: B. 60°.

So, the final answer to the question "What is the reference angle for a 240° angle?" is B. 60°. Great job, everyone! You've now mastered this key concept in trigonometry. Keep practicing, and you'll be solving these problems like a pro in no time! Remember, understanding reference angles is crucial for simplifying trigonometric calculations and building a solid foundation in trigonometry. Keep up the great work, and you'll be acing those exams in no time!