TRIGONOMETRYTrigonometryMathematics Calculator
tan

The Tangent Function

Tangent is the ratio of sine to cosine: tan(θ) = sin(θ)/cos(θ). It equals opposite/adjacent in a right triangle and represents the slope of a line at angle θ.

Concept Fundamentals
(-∞, ∞)
Range
π (180°)
Period
θ = π/2 + nπ
Asymptotes
tan²θ + 1 = sec²θ
Identity

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Tangent is positive in Q1 and Q3 (sin and cos same sign), negative in Q2 and Q4. tan(30°)=1/√3, tan(45°)=1, tan(60°)=√3 — the slope of a line at that angle. The inverse arctan returns angles in (-π/2, π/2); atan2 handles all quadrants.

Key quantities
(-∞, ∞)
Range
Key relation
π (180°)
Period
Key relation
θ = π/2 + nπ
Asymptotes
Key relation
tan²θ + 1 = sec²θ
Identity
Key relation

Ready to run the numbers?

Why: Tangent models slopes, elevation angles, and growth rates. Surveyors use it for distances; calculus uses it for derivatives: d/dx[tan(x)] = sec²(x).

How: tan(θ) = sin(θ)/cos(θ) = opposite/adjacent. When cos(θ)=0 (at 90°, 270°), tangent is undefined. The function has period π because both sin and cos flip sign.

Tangent is positive in Q1 and Q3 (sin and cos same sign), negative in Q2 and Q4.tan(30°)=1/√3, tan(45°)=1, tan(60°)=√3 — the slope of a line at that angle.
Sources:Wolfram MathWorld — TangentKhan Academy — Trigonometry

Run the calculator when you are ready.

Start CalculatingEnter an angle to compute tan(θ) — undefined at 90°, 270°

Examples — Click to Load

tangent.sh
CALCULATED
$ tan --angle 45° --all-functions
tan(θ)
1
sin(θ)
0.70710678
cos(θ)
0.70710678
Quadrant
Q1
csc(θ)
1.41421356
sec(θ)
1.41421356
cot(θ)
1
Ref Angle
45°
Share:
Tangent Calculator Result
tan(45°)
1
Q1ref 45°tan²+1 = 2
numbervibe.com/calculators/mathematics/trigonometry/tangent-calculator

Trig Value Breakdown

All 6 Trig Functions

tan² vs 1 (tan²+1=sec²)

Calculation Breakdown

CONVERSION
Input Angle
45°
Convert to Radians
0.78539816 rad
45° × π/180
Normalized Angle
45°
\text{theta} mod 360^{circ}
Quadrant
Q1
45.0° is in quadrant 1
Reference Angle
45°
ext{Acute} ext{angle} ext{to} x- ext{axis}
PRIMARY RESULT
TANGENT VALUE
1
tan(45°) = sin/cos
RELATED VALUES
Sine
0.70710678
sin(45°)
Cosine
0.70710678
cos(45°)
Cosecant
1.41421356
1/\text{sin}(\text{theta} )
Secant
1.41421356
1/\text{cos}(\text{theta} )
Cotangent
1
\text{cos}(\text{theta} )/\text{sin}(\text{theta} )
PYTHAGOREAN IDENTITY
tan²(θ)
1
tan²(θ) + 1
2
= ext{sec}^{2}(\text{theta} )
sin²(θ) + cos²(θ)
1
ext{Always} ext{equals} 1

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

Tangent literally means 'touching' — the tangent line to the unit circle at (1,0) extends to meet the terminal side.

— Wolfram MathWorld

∞

Unlike sine and cosine, tangent has no maximum — it approaches ±∞ at vertical asymptotes.

— Paul's Online Notes

Key Takeaways

  • • tan(θ) = opposite / adjacent in a right triangle; equivalently tan(θ) = sin(θ)/cos(θ)
  • • Tangent has vertical asymptotes at 90°, 270°, etc. (where cos = 0). Range: all real numbers
  • • Tangent is an odd function: tan(-θ) = -tan(θ) with period π (180°) — half of sine and cosine
  • • The identity tan²(θ) + 1 = sec²(θ) relates tangent to secant
  • • Tangent is positive in Q1 and Q3 (where sin and cos have same sign), negative in Q2 and Q4

Did You Know?

📐Tangent literally means 'touching' — the tangent line to the unit circle at (1,0) extends to meet the terminal side of angle θ. The length of that segment equals tan(θ)Source: Wolfram MathWorld
↗️Tangent equals slope: tan(θ) = rise/run. The angle a line makes with the positive x-axis has tangent equal to the line's slope. Essential for surveying and navigationSource: Khan Academy
∞Unlike sine and cosine, tangent has no maximum — it approaches ±∞ at asymptotes (90° + n×180°). The graph has distinct vertical "gaps"Source: Paul's Online Notes
🔄Tangent has period π, not 2π. The function repeats every 180° because tan(θ+π) = tan(θ). This comes from sin and cos both flipping signSource: MIT OpenCourseWare
🏔️Surveyors use tangent to calculate elevation angles and distances. The tangent of the angle of elevation gives the height-to-distance ratioSource: Engineering Toolbox
📡In calculus, d/dx[tan(x)] = sec²(x) = 1 + tan²(x). The derivative of tangent is always positive — tangent is strictly increasing between asymptotesSource: NIST DLMF

How the Tangent Function Works

Tangent is the ratio of sine to cosine: tan(θ) = sin(θ)/cos(θ). On the unit circle, it represents the length of the tangent line segment from (1,0) to the extended terminal side of angle θ.

Asymptotes and Undefined Values

When cos(θ) = 0 (at θ = 90°, 270°, etc.), the ratio sin/cos is undefined — division by zero. The tangent graph has vertical asymptotes at these angles. As θ approaches 90° from the left, tan(θ) → +∞; from the right, tan(θ) → -∞.

Period π (180°)

Tangent repeats every π radians because tan(θ+π) = sin(θ+π)/cos(θ+π) = (-sin θ)/(-cos θ) = tan θ. Both numerator and denominator flip sign, so the ratio stays the same. This is why the tangent graph has a shorter period than sine and cosine.

Slope Interpretation

If a line makes angle θ with the positive x-axis, its slope is tan(θ). A 45° line has slope 1; a 60° line has slope √3 ≈ 1.732. This geometric interpretation connects trigonometry to coordinate geometry.

Expert Tips

Memorize the Special Angles

tan(30°)=1/√3, tan(45°)=1, tan(60°)=√3. The pattern: 1/√3, 1, √3 for 30°, 45°, 60°. Use the Unit Circle Calculator to see the geometry.

Q1 and Q3 Are Positive

Tangent = sin/cos. Positive when sin and cos have the same sign (Q1 and Q3), negative when they differ (Q2 and Q4). Try the Sum & Difference Calculator.

Watch for Asymptotes

At 90°, 270°, and θ = π/2 + nπ, tangent is undefined. This calculator detects these and shows an error. In calculus, limits at these points are ±∞ depending on direction.

The tan²+1=sec² Identity

tan²θ + 1 = sec²θ. Divide sin²+cos²=1 by cos² to get it. Use this to find secant from tangent, or to integrate sec². See the Trig Identities Calculator.

Why Use This Calculator vs. Other Tools?

FeatureThis CalculatorScientific CalculatorManual Computation
All 6 trig functions at once✅❌ One at a time❌
Asymptote detection✅❌ Shows error⚠️ Manual
Quadrant & reference angle✅❌✅ Slow
Visual charts & breakdown✅❌❌
Step-by-step explanation✅❌✅
tan²+1=sec² verification✅❌⚠️ Manual
Copy & share results✅❌❌
Preset examples✅❌❌

Frequently Asked Questions

When is tangent undefined?

Tangent is undefined at 90°, 270°, and all angles where cos(θ) = 0 — that is, θ = π/2 + nπ (or 90° + n×180°). At these angles, the terminal side is vertical, and tan = sin/cos involves division by zero. The graph has vertical asymptotes at these points.

What is the period of tangent?

The period of tan(θ) is π radians (180°), half that of sine and cosine. So tan(θ + π) = tan(θ). This is because both sin and cos change sign when adding π, so their ratio stays the same.

Why is tangent positive in Q1 and Q3?

Tangent = sin/cos. In Q1, both sin and cos are positive. In Q3, both are negative. A negative divided by a negative is positive. In Q2 and Q4, one is positive and one negative, so the ratio is negative.

What is tan²(θ) + 1?

tan²(θ) + 1 = sec²(θ). This identity comes from dividing sin²+cos²=1 by cos². It is essential for calculus (integrating sec²) and for finding secant from tangent.

How does tangent relate to slope?

If a line makes angle θ with the positive x-axis, its slope is tan(θ). So a 45° line has slope 1, a 60° line has slope √3. This is because slope = rise/run = opposite/adjacent in the right triangle formed.

What is the range of tangent?

Tangent can take any real value: (-∞, +∞). Unlike sine and cosine, it is unbounded. As θ approaches 90° from the left, tan(θ) → +∞; from the right, tan(θ) → -∞.

Why is tangent called an odd function?

tan(-θ) = -tan(θ). A function is odd when f(-x) = -f(x). Graphically, the tangent curve has rotational symmetry about the origin — rotating 180° gives the same graph.

Where is tangent used in real life?

Tangent appears in surveying (elevation angles, distances), navigation (bearing calculations), physics (inclined planes, friction), engineering (slope stability, ramp design), and computer graphics (perspective, 3D rotations).

Tangent Function by the Numbers

(-∞, ∞)
Output Range
180°
Period
π/2
Asymptotes
Slope
Geometric Meaning

Disclaimer: This calculator provides results based on standard IEEE 754 floating-point arithmetic. Results are accurate to approximately 15 significant digits. Tangent is undefined at 90°, 270°, etc. — the calculator will display an error at these angles. Not a substitute for professional engineering analysis.

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