Finding the Obtuse Angle Between Lines Using Slopes

This question asks us to find the obtuse angle between two lines, given their slopes. The slopes of the two lines are provided as \(m_1 = 2 - \sqrt{3}\) and \(m_2 = 2 + \sqrt{3}\). To find the angle between two lines, we can use a standard formula that relates the slopes to the tangent of the angle.

Formula for the Angle Between Two Lines

The formula for the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is given by:

\(\tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|\)

This formula gives the acute angle between the lines. If we need the obtuse angle, we can find the acute angle first and then subtract it from \(180^\circ\).

Calculating the Terms for the Formula

We are given the slopes:

• Slope 1, \(m_1 = 2 - \sqrt{3}\)
• Slope 2, \(m_2 = 2 + \sqrt{3}\)

Let's calculate the numerator term, \(m_2 - m_1\):

\(m_2 - m_1 = (2 + \sqrt{3}) - (2 - \sqrt{3}) = 2 + \sqrt{3} - 2 + \sqrt{3} = 2\sqrt{3}\)

Now, let's calculate the denominator term, \(1 + m_1 m_2\). First, find the product \(m_1 m_2\):

\(m_1 m_2 = (2 - \sqrt{3})(2 + \sqrt{3})\)

This product is in the form \((a-b)(a+b)\), which simplifies to \(a^2 - b^2\). Here, \(a=2\) and \(b=\sqrt{3}\).

\(m_1 m_2 = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1\)

Now substitute this back into the denominator term:

\(1 + m_1 m_2 = 1 + 1 = 2\)

Calculating the Tangent of the Angle

Now we can substitute the calculated numerator and denominator values into the formula for \(\tan \theta\):

\(\tan \theta = \left| \frac{2\sqrt{3}}{2} \right|\)

Simplify the expression inside the absolute value:

\(\tan \theta = \left| \sqrt{3} \right|\)

\(\tan \theta = \sqrt{3}\)

Determining the Angles

We need to find the angle \(\theta\) whose tangent is \(\sqrt{3}\). We know from trigonometry that \(\tan 60^\circ = \sqrt{3}\). Therefore, the acute angle between the two lines is \(60^\circ\).

The question asks for the obtuse angle between the lines. The acute and obtuse angles between two lines are supplementary, meaning they add up to \(180^\circ\).

Obtuse Angle = \(180^\circ - \text{Acute Angle}\)

Obtuse Angle = \(180^\circ - 60^\circ = 120^\circ\)

So, the obtuse angle between the lines with slopes \(2 - \sqrt{3}\) and \(2 + \sqrt{3}\) is \(120^\circ\).

Conclusion

The obtuse angle between the lines is \(120^\circ\). This matches one of the given options.

Revision Table: Angle Between Lines

Additional Information on Line Angles and Slopes

Understanding the relationship between the slopes of lines and the angle between them is fundamental in coordinate geometry. The slopes \(m_1 = 2 - \sqrt{3}\) and \(m_2 = 2 + \sqrt{3}\) used in this problem are related to standard trigonometric values.

• The value \(\sqrt{3}\) is the tangent of \(60^\circ\) or \(\pi/3\) radians.
• The value \(1/\sqrt{3}\) is the tangent of \(30^\circ\) or \(\pi/6\) radians.
• The numbers \(2 - \sqrt{3}\) and \(2 + \sqrt{3}\) are related to \(\tan 15^\circ\) and \(\tan 75^\circ\) respectively, or cotangents. Specifically, \(\tan 15^\circ = 2 - \sqrt{3}\) and \(\tan 75^\circ = 2 + \sqrt{3}\).

If the slopes were \(\tan \alpha\) and \(\tan \beta\), the angle \(\theta\) between the lines would satisfy \(\tan \theta = \left| \frac{\tan \beta - \tan \alpha}{1 + \tan \alpha \tan \beta} \right| = |\tan(\beta - \alpha)|\). In our case, if we let \(m_1 = \tan \alpha = 2 - \sqrt{3}\) and \(m_2 = \tan \beta = 2 + \sqrt{3}\), then \(\alpha = 15^\circ\) and \(\beta = 75^\circ\). The difference \(\beta - \alpha = 75^\circ - 15^\circ = 60^\circ\). The tangent of the absolute difference is \(\tan(60^\circ) = \sqrt{3}\), which matches our calculation using the formula. The angle between the lines is \(| \beta - \alpha |\) or \(180^\circ - | \beta - \alpha |\). Since the acute angle is \(60^\circ\), the obtuse angle is \(180^\circ - 60^\circ = 120^\circ\). This confirms the result obtained directly from the slope formula.