1. What is Small Angle Approximation?

For very small angles (θ < 0.1 radians or about 5.7°), the angle in radians is approximately equal to its tangent: θ ≈ tan θ. This approximation is useful when calculators are not available.

2. How Does This Calculator Work?

The calculator uses the small angle approximation:

Where:

• θ — Angle in radians
• tan θ — Tangent of the angle

Explanation: For angles less than about 0.1 radians, the difference between θ and tan θ is less than 0.17%.

3. When is This Approximation Valid?

Details: This approximation works best for angles below 0.1 radians (5.7°). The error grows rapidly for larger angles.

4. Using the Calculator

Tips: Enter the tangent value (opposite/adjacent ratio) of a small angle. The calculator will return the angle in both radians and degrees.

5. Frequently Asked Questions (FAQ)

Q1: Why does this approximation work?
A: The Taylor series expansion of tan θ shows that for small θ, tan θ ≈ θ + (θ³/3) + ..., so the first term dominates.

Q2: How accurate is this approximation?
A: For θ = 0.1 rad (5.7°), the error is about 0.17%. At θ = 0.2 rad (11.5°), error grows to about 1.4%.

Q3: What are some practical applications?
A: Useful in optics (small angle approximations), astronomy (angular sizes), and engineering calculations without calculators.

Q4: When should I not use this approximation?
A: When the angle is larger than about 0.2 radians (11.5°) or when high precision is required.

Q5: How can I convert between degrees and radians?
A: 1 radian = 180/π degrees ≈ 57.2958°. The calculator shows both units.