Comparing Angles: Lines with Slopes 2 and 1/2 on X-Axis

Q: Why does a slope of 2 make a bigger angle than 1/2?

Think of slope as rise over run! A slope of 2 means going up 2 units for every 1 unit right - that's steep. A slope of 1/2 means going up only 1/2 unit for every 1 unit right - that's gentle.

Q: Do I need to calculate the actual angle values?

Not usually! Since $ \tan^{-1}(x) $ is an increasing function, you can just compare the slopes directly. Bigger slope = bigger angle.

Q: What if one slope is negative?

Negative slopes create negative angles (below the x-axis). When comparing with positive slopes, the positive slope always creates the larger angle with the x-axis.

Q: How can I visualize this without a calculator?

Draw quick sketches! A line with slope 2 is steep (closer to vertical), while slope 1/2 is gentle (closer to horizontal). The steeper line clearly makes a bigger angle.

Slope-Angle Relationships with Arctangent Functions

Two straight lines have slopes of $2,\frac{1}{2}$ .

Which of the lines forms a larger angle with the x axis?

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's find which line has a larger angle with the X-axis.

00:12 We'll use a formula to find the slope based on the angle with the X-axis. Ready?

00:18 Substitute the slope with the given data. Then, calculate to find the angle.

00:22 Next, we'll separate out the angle.

00:25 And here is the angle of the first line!

00:31 Now, let's do the same for the second line.

00:35 Substitute the slope again using the data given. Then, calculate for the angle.

00:40 Once more, isolate the angle.

00:46 And here is the angle of the second line!

00:52 That's how you solve this question. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Two straight lines have slopes of $2,\frac{1}{2}$ .

Which of the lines forms a larger angle with the x axis?

Step-by-step solution

To solve for which line forms a larger angle with the x-axis, we'll proceed by calculating the arctangent of each slope:

First, calculate the angle for the line with slope $m_1 = 2$ :

$\theta_1 = \tan^{-1}(2)$

Next, calculate the angle for the line with slope $m_2 = \frac{1}{2}$ :

$\theta_2 = \tan^{-1}\left(\frac{1}{2}\right)$

Comparing these two results:

The function $\tan^{-1}(x)$ is increasing, meaning as the value of $x$ increases, so does the angle $\theta$ . Therefore, since $2 > \frac{1}{2}$ , it follows that $\theta_1 > \theta_2$ .

Therefore, the straight line with a slope of 2 forms the largest angle with the x-axis.

Final Answer

The straight line with a slope of 2 forms the largest angle.

Key Points to Remember

Essential concepts to master this topic

Rule: Larger slopes create larger angles with x-axis
Technique: Use $\theta = \tan^{-1}(m)$ where $\tan^{-1}(2) > \tan^{-1}(\frac{1}{2})$
Check: Since arctangent is increasing, compare slopes directly: 2 > 1/2 ✓

Common Mistakes

Avoid these frequent errors

Thinking smaller slopes create larger angles
Don't assume that 1/2 makes a bigger angle than 2 because fractions seem "bigger"! This confuses the reciprocal relationship. Larger slope values always create steeper lines with larger angles from the x-axis. Always remember: steeper slope = larger angle.

Practice Quiz

Test your knowledge with interactive questions

Look at the linear function represented in the diagram.

When is the function positive?

FAQ

Everything you need to know about this question

Why does a slope of 2 make a bigger angle than 1/2?

Think of slope as rise over run! A slope of 2 means going up 2 units for every 1 unit right - that's steep. A slope of 1/2 means going up only 1/2 unit for every 1 unit right - that's gentle.

Do I need to calculate the actual angle values?

Not usually! Since $\tan^{-1}(x)$ is an increasing function, you can just compare the slopes directly. Bigger slope = bigger angle.

What if one slope is negative?

Negative slopes create negative angles (below the x-axis). When comparing with positive slopes, the positive slope always creates the larger angle with the x-axis.

How can I visualize this without a calculator?

Draw quick sketches! A line with slope 2 is steep (closer to vertical), while slope 1/2 is gentle (closer to horizontal). The steeper line clearly makes a bigger angle.

Is there a maximum angle a line can make?

Yes! As the slope approaches infinity, the angle approaches 90 degrees (vertical line). As slope approaches zero, the angle approaches 0 degrees (horizontal line).

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Linear Functions questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime