MATHEMATICSTrigonometryMathematics Calculator
🔢

Secant

Calculate the secant of any angle in degrees or radians. Get all 6 trig functions, quadrant info, reference angle, sec²=1+tan² identity — with interactive charts and step-by-step breakdown.

Did our AI summary help? Let us know.

Why: Understanding secant helps you make better, data-driven decisions.

How: Enter Angle, Unit to calculate results.

Run the calculator when you are ready.

Start CalculatingExplore mathematical calculations

Examples — Click to Load

secant.sh
CALCULATED
$ sec --angle 0° --all-functions
sec(θ)
1
cos(θ)
1
tan(θ)
0
Quadrant
Q1
sin(θ)
0
csc(θ)
undef
cot(θ)
undef
Ref Angle
0°
Share:
Secant Calculator Result
sec(0°)
1
Q1ref 0°sec² = 1
numbervibe.com/calculators/mathematics/trigonometry/secant-calculator

Trig Value Breakdown

All 6 Trig Functions

sin² vs cos² (Pythagorean Identity)

Calculation Breakdown

CONVERSION
Input Angle
0°
Convert to Radians
0 rad
0° × π/180
Normalized Angle
0°
\text{theta} mod 360^{circ}
Quadrant
Q1
0.0° is in quadrant 1
Reference Angle
0°
ext{Acute} ext{angle} ext{to} x- ext{axis}
PRIMARY RESULT
SECANT VALUE
1
sec(0°) = 1/cos(θ)
RELATED VALUES
Cosine
1
cos(0°)
Sine
0
sin(0°)
Tangent
0
tan(0°) = sin/cos
Cosecant
undefined
1/\text{sin}(\text{theta} )
Cotangent
undefined
\text{cos}(\text{theta} )/\text{sin}(\text{theta} )
IDENTITY
sec²(θ)
1
1 + tan²(θ)
1
ext{sec}^{2} = 1 + \text{tan}^{2}

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

Key Takeaways

  • • sec(θ) = 1/cos(θ) = hypotenuse/adjacent in a right triangle; secant is the reciprocal of cosine
  • • Secant is undefined at 90° and 270° where cos(θ) = 0. Range: (-∞,-1] ∪ [1,∞) — never between -1 and 1
  • • Secant is an even function: sec(-θ) = sec(θ) with period 2π (360°)
  • • The Pythagorean identity sec²(θ) = 1 + tan²(θ) always holds when sec is defined
  • • Secant has the same sign as cosine: positive in Q1 and Q4, negative in Q2 and Q3

Did You Know?

✂️The name 'secant' comes from Latin 'secare' meaning 'to cut' — the secant line cuts through the unit circle, extending beyond itSource: Wolfram MathWorld
📐In surveying and navigation, secant appears in formulas for calculating distances when angles are near 90°Source: Khan Academy
🔬Secant is used in optics for Snell's law calculations when light passes between media at grazing anglesSource: MIT OpenCourseWare
📡Antenna design and signal propagation use secant in beam pattern calculations and Fresnel zone analysisSource: IEEE Antennas
🏛️Architects use secant in structural calculations for arches and domes where load distribution follows curved pathsSource: Engineering Toolbox
🎯Secant method is a numerical root-finding algorithm — a faster alternative to the bisection method in calculusSource: Paul's Online Notes

How the Secant Function Works

The secant function is the reciprocal of cosine: sec(θ) = 1/cos(θ). On the unit circle, when the x-coordinate (cosine) is small, secant becomes large in magnitude.

Where Secant is Undefined

At 90° and 270°, the terminal side is vertical, so cos(θ) = 0. Division by zero makes sec(θ) undefined. The secant graph has vertical asymptotes at these angles, approaching ±∞ from either side.

Range and Even Symmetry

Since |cos(θ)| ≤ 1, we have |sec(θ)| ≥ 1. Secant never lies between -1 and 1. As an even function, sec(-θ) = sec(θ), so the graph is symmetric about the y-axis.

sec² = 1 + tan² Identity

Dividing sin²θ + cos²θ = 1 by cos²θ gives 1 + tan²θ = sec²θ. This identity connects secant to tangent and is essential for integration and solving trig equations.

Expert Tips

Memorize Special Angles

sec(0°)=1, sec(30°)=2/√3≈1.155, sec(45°)=√2≈1.414, sec(60°)=2. Use the Unit Circle Calculator to visualize.

Avoid 90° and 270°

Always check if your angle is near 90° or 270° before computing sec. Use the Cosine Calculator to verify cos ≠ 0.

sec² = 1 + tan²

If you know tan(θ), then sec(θ) = ±√(1 + tan²θ). The sign matches cos(θ). See the Trig Identities Calculator.

Reciprocal Relationship

sec(θ) · cos(θ) = 1 always. When cos is small, sec is large. When cos = 1 (at 0°), sec = 1. Compare with Cosecant (1/sin).

Why Use This Calculator vs. Other Tools?

FeatureThis CalculatorScientific CalculatorManual Computation
All 6 trig functions at once✅❌ One at a time❌
Undefined detection (90°, 270°)✅⚠️ May show error✅ Slow
Visual charts & breakdown✅❌❌
Step-by-step explanation✅❌✅
sec² = 1 + tan² identity check✅❌⚠️ Manual
Copy & share results✅❌❌
Degrees and radians✅✅✅
Preset examples✅❌❌

Frequently Asked Questions

When is secant undefined?

Secant is undefined at 90°, 270°, and all angles where cos(θ) = 0. At these angles, the terminal side is vertical on the unit circle, so the x-coordinate (cosine) is zero and division by zero occurs.

What is the range of secant?

The range of sec(θ) is (-∞,-1] ∪ [1,∞). Secant never outputs values between -1 and 1 because |cos(θ)| ≤ 1, so |sec(θ)| = 1/|cos(θ)| ≥ 1.

Why is secant called an even function?

A function is even when f(-x) = f(x). For secant: sec(-θ) = 1/cos(-θ) = 1/cos(θ) = sec(θ) because cosine is even. The secant graph is symmetric about the y-axis.

How do I find secant without a calculator?

First find cos(θ) using special angles or reference angles. Then sec(θ) = 1/cos(θ). Memorize: sec(0°)=1, sec(30°)=2/√3, sec(45°)=√2, sec(60°)=2. Avoid 90° and 270°.

What is sec²(θ) - tan²(θ)?

Always equals 1. From the identity sec²θ = 1 + tan²θ, we get sec²θ - tan²θ = 1. This is used extensively in calculus for integration (e.g., ∫sec²x dx = tan x + C).

Where is secant used in real life?

Secant appears in optics (Snell's law at grazing angles), surveying, antenna design, structural engineering for arches, and the secant method for numerical root-finding in computer science.

What is the period of secant?

The period of sec(θ) is 2π radians (360°), same as cosine. sec(θ + 2π) = sec(θ). The graph repeats every full rotation around the unit circle.

How is secant related to cosine?

Secant is the reciprocal of cosine: sec(θ) = 1/cos(θ). They have the same sign. When cos is near 0, sec approaches ±∞. When cos = 1, sec = 1.

Secant Function by the Numbers

(-∞,-1]∪[1,∞)
Output Range
360°
Period
2
Undefined Angles
Even
Symmetry

Disclaimer: This calculator provides results based on standard IEEE 754 floating-point arithmetic. Results are accurate to approximately 15 significant digits. Secant is undefined at 90° and 270° — the calculator will display an error for these inputs. For mission-critical applications, always verify with certified computational tools.

🚀 DIVING IN
🏊Let's explore the numbers!
AI

Related Calculators

Cosecant Calculator

Calculate the cosecant of any angle in degrees or radians. Get csc(θ) = 1/sin(θ), undefined detection at 0°/180°, step-by-step solutions, and interactive...

Mathematics

Cosine Calculator

Calculate the cosine of any angle in degrees or radians. Get cos(θ), related sin and tan values, quadrant info, step-by-step solutions, and interactive...

Mathematics

Sine Calculator

Calculate the sine of any angle in degrees or radians. Get sin(θ), related cos and tan values, quadrant info, step-by-step solutions, and interactive charts....

Mathematics

Tangent Calculator

Calculate the tangent of any angle in degrees or radians. Get tan(θ), related sin and cos values, quadrant info, asymptote detection, step-by-step solutions...

Mathematics

Cotangent Calculator

Calculate the cotangent of any angle in degrees or radians. Get cot(θ) = cos(θ)/sin(θ), undefined detection at 0° and 180°, step-by-step solutions, and...

Mathematics

Unit Circle Calculator

Find x (cos), y (sin), reference angle, quadrant for any angle. Step-by-step, charts.

Mathematics