Calculate the inclination angle of a line given its slope using inverse tangent and angle calculation | Step-by-Step Solution

🌟 Understanding Line Inclination from Slope

1. What We're Solving:

We need to find the inclination angle (the angle a line makes with the positive x-axis, measured counterclockwise) when we know the line has a slope of -1/√3.

2. The Approach:

The key relationship is that slope = tan(θ), where θ is the inclination angle. Since we know the slope, we can use the inverse tangent (arctan) to find the angle. We need to be careful about which quadrant our angle should be in, since slopes can be negative!

3. Step-by-Step Solution:

Step 1: Set up the relationship

• We know: slope = -1/√3
• We use: slope = tan(θ)
• So: tan(θ) = -1/√3

Step 2: Find the reference angle

• First, find what angle has tangent = 1/√3 (the positive version)
• tan⁻¹(1/√3) = 30° (or π/6 radians)
• This is our reference angle!

Step 3: Determine the correct quadrant

• Since our slope is NEGATIVE (-1/√3), our line is falling from left to right
• For inclination angles (measured counterclockwise from positive x-axis), negative slopes correspond to angles in the second quadrant (between 90° and 180°)

Step 4: Calculate the final angle

• In the second quadrant: θ = 180° - 30° = 150°
• Or in radians: θ = π - π/6 = 5π/6 radians

Step 5: Verify our answer

• Let's check: tan(150°) = tan(180° - 30°) = -tan(30°) = -1/√3 ✓
• Perfect! This matches our given slope.

4. The Answer:

The inclination angle is 150° (or 5π/6 radians).

5. Memory Tip:

Remember the "Slope-Angle Dance":

• Positive slope → angle between 0° and 90° (line going upward)
• Negative slope → angle between 90° and 180° (line going downward)
• The reference angle (30° in this case) helps you find exactly where in that range you land!

Great job working through this! The connection between slope and tangent is one of the most beautiful relationships in coordinate geometry. Keep practicing, and these angle calculations will become second nature! 🎯