Compare Lines with Slopes -3 and -6: Finding the Greater X-Axis Angle

Q: Why does the line with slope -6 have a smaller angle than slope -3?

Think about it visually! A line with slope -6 is steeper (more vertical) than slope -3. When we measure the positive angle from the x-axis, steeper negative slopes create smaller acute angles.

Q: How do I calculate the actual angle values?

Use your calculator: $ \tan^{-1}(-3) ≈ -71.57° $ and $ \tan^{-1}(-6) ≈ -80.54° $. The absolute values show that 71.57° > 80.54° is false, so 71.57° is actually the larger positive angle!

Q: What does 'angle with the x-axis' really mean?

It's the acute angle between the line and the positive x-axis, measured counterclockwise. For negative slopes, we consider the positive equivalent of this angle.

Q: Can I just compare the slope numbers directly?

No! You must use the inverse tangent function or think about absolute values. Comparing -3 and -6 directly gives the wrong relationship for angles.

Q: Why do we use absolute values when comparing?

Because we're looking at the magnitude of the angle, not its direction. The absolute value \( |-3| = 3 , but this means slope -3 creates a larger positive angle with the x-axis.

Line Slopes with Angle Comparison

Two lines haves slopes of -3 and -6.

Which of the lines forms a greater 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 Let's substitute the slope according to the given data and calculate to find the angle

00:11 Let's isolate the angle

00:24 This is the angle of the first line

00:30 Let's calculate the angle of the line with the X-axis

00:35 This is its angle with the X-axis

00:39 We'll use the same method to find the angle of the second line

00:45 Let's substitute the slope according to the given data and calculate to find the angle

00:48 Let's isolate the angle

00:55 This is the angle of the second line

01:03 Let's calculate the angle of the line with the X-axis

01:10 This is its angle with the X-axis

01:13 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 lines haves slopes of -3 and -6.

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

Step-by-step solution

To find which line forms a greater angle with the x-axis, we use the formula $\theta = \tan^{-1}(m)$ . This gives us the angles formed by lines with slopes $m_1 = -3$ and $m_2 = -6$ .

We first calculate the inverse tangent for both slopes:

For $m_1 = -3$ , the angle is $\theta_1 = \tan^{-1}(-3)$ .
For $m_2 = -6$ , the angle is $\theta_2 = \tan^{-1}(-6)$ .

The greater the absolute value of the negative slope, the closer the line is to the vertical, thus forming a greater angle with the x-axis.

Comparing $|-3|$ and $|-6|$ , we see that $-6$ has a greater absolute value, indicating a steeper angle. Hence, although $-6$ is steeper, it forms a smaller angle with the x-axis because we're considering angles formed in the positive x-direction (i.e., above the x-axis).

Therefore, the line whose slope is $-3$ forms the greater angle with the x-axis.

The line whose slope is -3 forms the greater angle.

Final Answer

The line whose slope is -3 forms the greater angle.

Key Points to Remember

Essential concepts to master this topic

Angle Formula: Use $\theta = \tan^{-1}(m)$ to find angle with x-axis
Technique: Compare absolute values: |-3| = 3 vs |-6| = 6
Check: Steeper slope means greater absolute value but smaller positive angle ✓

Common Mistakes

Avoid these frequent errors

Thinking steeper slope always means greater angle
Don't assume that slope -6 gives a greater angle than slope -3 = wrong answer! Steeper negative slopes actually create smaller positive angles with the x-axis. Always remember that as negative slopes get steeper, their positive angles with the x-axis decrease.

Practice Quiz

Test your knowledge with interactive questions

Look at the function shown in the figure.

When is the function positive?

FAQ

Everything you need to know about this question

Why does the line with slope -6 have a smaller angle than slope -3?

Think about it visually! A line with slope -6 is steeper (more vertical) than slope -3. When we measure the positive angle from the x-axis, steeper negative slopes create smaller acute angles.

How do I calculate the actual angle values?

Use your calculator: $\tan^{-1}(-3) ≈ -71.57°$ and $\tan^{-1}(-6) ≈ -80.54°$ . The absolute values show that 71.57° > 80.54° is false, so 71.57° is actually the larger positive angle!

What does 'angle with the x-axis' really mean?

It's the acute angle between the line and the positive x-axis, measured counterclockwise. For negative slopes, we consider the positive equivalent of this angle.

Can I just compare the slope numbers directly?

No! You must use the inverse tangent function or think about absolute values. Comparing -3 and -6 directly gives the wrong relationship for angles.

Why do we use absolute values when comparing?

Because we're looking at the magnitude of the angle, not its direction. The absolute value $|-3| = 3 < 6 = |-6|$ , but this means slope -3 creates a larger positive angle with the x-axis.

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