Understanding Tangent in Trigonometry

The tangent function is a fundamental trigonometric ratio that relates the angles of a right triangle to the ratio of its opposite and adjacent sides. It's widely used in mathematics, physics, engineering, and navigation.

Key Formulas

tan(θ) = Opposite / Adjacent
θ = arctan(Opposite / Adjacent)
Angle conversion: radians = degrees × π/180

Practical Examples

Example 1: For a 45° angle, tan(45°) = 1

Example 2: With opposite side = 3 and adjacent side = 4, tan(θ) = 0.75 and θ ≈ 36.87°

Example 3: For a 60° angle, tan(60°) = √3 ≈ 1.732

Common Tangent Values

Angle (°)	tan(θ)
0	0
30	1/√3 ≈ 0.577
45	1
60	√3 ≈ 1.732
90	Undefined

Important Notes

Tangent is undefined at 90° and 270° (π/2 and 3π/2 radians)
The tangent function has a period of 180° (π radians)
For angles greater than 90°, consider the quadrant for proper sign determination
Results are approximate due to floating-point precision limitations

Frequently Asked Questions

Why is tangent undefined at 90°?

At 90°, the adjacent side becomes zero, making the ratio opposite/adjacent undefined (division by zero).

How accurate are the calculations?

Results are accurate to 10 decimal places, but JavaScript floating-point precision may cause minor rounding errors in extreme cases.

Can I calculate tangent for angles beyond 90°?

Yes, but our calculator uses the principal values between -90° and 90°. For other angles, use periodicity: tan(θ) = tan(θ ± 180°).

Tangent Calculator

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Tangent Calculator - tan Angle or Opposite over Adjacent

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