Tangent Calculator (tan)
Calculate the tangent of any angle in degrees or radians. Identifies vertical asymptotes, exact values for special angles, and ratios.
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What Is a Tangent Calculator (tan)?
In trigonometry, the tangent of an angle is the ratio of the opposite leg to the adjacent leg in a right triangle: tan(θ) = Opposite / Adjacent = sin(θ) / cos(θ). It also represents the slope of the terminal ray.
How to Use This Calculator
- Enter the angle θ and choose Degrees or Radians.
- View the decimal tangent value or undefined asymptote warning.
- Review the exact fractional value and step-by-step deduction.
Tangent Trigonometric Definition
\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin(\theta)}{\cos(\theta)} = \text{slope } mTangent is undefined whenever cos(θ) = 0, which occurs at odd multiples of 90° (±90°, ±270°, etc.).
Worked Example
Scenario: Calculate tan(45°)
Given: θ = 45°
In a 45°-45°-90° triangle, opposite leg equals adjacent leg.
tan(45°) = sin(45°) / cos(45°) = (√2/2) / (√2/2) = 1
Tips & Key Notes
- Tangent has vertical asymptotes at 90° + k·180° because dividing by zero is undefined.
- The period of the tangent function is 180° (or π radians), unlike sine and cosine which repeat every 360°.
- The tangent of an angle equals the slope (rise over run) of the line passing through that angle.
Frequently Asked Questions
Why is tan(90°) undefined?
At 90°, cos(90°) = 0. Because tan(θ) = sin(θ) / cos(θ), evaluating tan(90°) requires dividing by zero (1 / 0), which is mathematically undefined.
What are common exact tangent values?
tan(0°) = 0, tan(30°) = √3/3 ≈ 0.577, tan(45°) = 1, tan(60°) = √3 ≈ 1.732.
What is the period of the tangent function?
The period of tangent is 180° (or π radians): tan(θ + π) = tan(θ).
What is the reciprocal of tangent?
The reciprocal of tangent is cotangent: cot(θ) = 1 / tan(θ) = cos(θ) / sin(θ).
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