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?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find which of the lines has a larger angle with the X-axis

00:03 We'll use the formula to calculate slope based on the angle with the X-axis

00:07 We'll substitute the slope according to the given data and calculate to find the angle

00:12 We'll isolate the angle

00:18 This is the angle of the first line

00:24 We'll use the same method to find the angle of the second line

00:28 We'll substitute the slope according to the given data and calculate to find the angle

00:33 We'll isolate the angle

00:39 This is the angle of the second line

00:45 And this is the solution to the question

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

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Look at the function shown in the figure.

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).

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