Comparing Line Slopes: Finding Smaller Angle Between -6 and 1/2 Slopes

Q: Why does the slope of -6 give a negative angle?

The arctangent function gives angles between -90° and 90°. When slope is negative, the line goes downward from left to right, so the angle is negative. But we need the reference angle with the x-axis!

Q: How do I convert a negative angle to find the actual angle with x-axis?

For negative results from $ \tan^{-1} $, add 180° to get the reference angle. For example: $ \tan^{-1}(-6) = -80.53° $, so the angle is 180° - 80.53° = 99.47°.

Q: Which angle is smaller: 99.47° or 26.56°?

Clearly 26.56° is smaller than 99.47°. The line with slope $ \frac{1}{2} $ makes the smaller angle with the x-axis because it's closer to being horizontal.

Q: Can I just compare the slopes directly without finding angles?

No! You can't compare slopes directly for angles. A slope of -6 has a larger absolute value than $ \frac{1}{2} $, but it doesn't make the smaller angle. Always convert to actual angles first.

Q: What if both slopes were positive?

If both slopes are positive, the smaller slope makes the smaller angle. This is because as slope increases from 0, the line gets steeper and the angle with x-axis increases toward 90°.

Line Slope Angles with Arctangent Relationships

Two lines have slopes of $-6$ and $\frac{1}{2}$ .

Which of the lines forms a smaller angle with the x-axis?

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Step-by-step video solution

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

00:03 Use the formula to calculate slope based on angle with X-axis

00:07 Substitute the slope according to given data and calculate the angle

00:11 Isolate the angle

00:18 This is the angle of the first line

00:21 Find the angle of the line relative to X-axis

00:26 This is the angle relative to X-axis

00:31 Use the same method to find the angle of the second line

00:36 Substitute the slope according to given data and calculate the angle

00:40 Isolate the angle

00:45 This is the angle of the second line

00:50 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 have slopes of $-6$ and $\frac{1}{2}$ .

Which of the lines forms a smaller angle with the x-axis?

Step-by-step solution

We will use the formula:

$m=\tan\alpha$

Let's check the slope of minus 6:

$-6=\tan\alpha$

$\tan^{-1}(-6)=\alpha$

$-80.53=\alpha$

$180-80.53=$

$99.47=\alpha_1$

Let's check the slope of one-half:

$\frac{1}{2}=\tan\alpha$

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

$26.56=\alpha_2$

$\alpha_1 > \alpha_2$

Final Answer

The line with a slope of $\frac{1}{2}$

Key Points to Remember

Essential concepts to master this topic

Relationship: Line slope equals tangent of angle with x-axis
Method: Use $\alpha = \tan^{-1}(m)$ to find angle from slope
Check: Smallest absolute angle value gives smallest angle with x-axis ✓

Common Mistakes

Avoid these frequent errors

Using negative angles directly without finding reference angle
Don't use -80.53° as the final angle = wrong comparison! Negative angles from arctangent need conversion to positive reference angles between 0° and 180°. Always find the acute reference angle by taking 180° - |negative result|.

Practice Quiz

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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 the slope of -6 give a negative angle?

The arctangent function gives angles between -90° and 90°. When slope is negative, the line goes downward from left to right, so the angle is negative. But we need the reference angle with the x-axis!

How do I convert a negative angle to find the actual angle with x-axis?

For negative results from $\tan^{-1}$ , add 180° to get the reference angle. For example: $\tan^{-1}(-6) = -80.53°$ , so the angle is 180° - 80.53° = 99.47°.

Which angle is smaller: 99.47° or 26.56°?

Clearly 26.56° is smaller than 99.47°. The line with slope $\frac{1}{2}$ makes the smaller angle with the x-axis because it's closer to being horizontal.

Can I just compare the slopes directly without finding angles?

No! You can't compare slopes directly for angles. A slope of -6 has a larger absolute value than $\frac{1}{2}$ , but it doesn't make the smaller angle. Always convert to actual angles first.

What if both slopes were positive?

If both slopes are positive, the smaller slope makes the smaller angle. This is because as slope increases from 0, the line gets steeper and the angle with x-axis increases toward 90°.

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