Secant Calculator
Enter an angle in degrees or radians to find its secant, sec θ.
Updated
Secant Calculator
Our Secant Calculator finds sec θ for any angle in degrees or radians. Type the angle and the answer appears straight away to 10 decimal places, with the exact value for special angles (such as √2 for 45°), the angle in the other unit, the quadrant, and cos θ and sin θ. It tells you clearly when the secant is undefined, at 90°, 270° and similar angles, and gives sec 60° as exactly 2 rather than the 1.9999999999999996 many programs show.
Quick answer
The secant of an angle is the reciprocal of the cosine: 1 ÷ cos θ, or the hypotenuse divided by the adjacent side. For example, sec 60° = 1 ÷ 0.5 = 2. Enter any angle above and the calculator shows sec θ instantly.
- Formula: sec θ = 1 ÷ cos θ = hypotenuse ÷ adjacent
- Key values: sec 0° = 1, sec 30° ≈ 1.1547, sec 45° ≈ 1.4142, sec 60° = 2
- Undefined: at 90°, 270° and every 90° + k × 180°, where cos θ = 0
- Range and period: sec θ ≤ −1 or sec θ ≥ 1, repeating every 360° (2π)
What Is Secant?
Secant is one of the three reciprocal trigonometric functions, with cosecant and cotangent. It flips the cosine upside down, so in a right triangle it compares the hypotenuse with the side next to the angle:
sec θ = 1 ÷ cos θ
cos θ = adjacent ÷ hypotenuse
Because cosine never goes beyond −1 or 1, the secant never goes between them: dividing 1 by a number smaller than 1 always gives something larger than 1. Where cos θ is 0, the secant has no value at all.
Secant Table of Common Angles
| Angle | Exact | Decimal |
|---|---|---|
| 0° (0) | 1 | 1 |
| 30° (π/6) | 2√3/3 | 1.1547 |
| 45° (π/4) | √2 | 1.4142 |
| 60° (π/3) | 2 | 2 |
| 90° (π/2) | undefined | undefined |
| 120° (2π/3) | −2 | −2 |
| 135° (3π/4) | −√2 | −1.4142 |
| 150° (5π/6) | −2√3/3 | −1.1547 |
| 180° (π) | −1 | −1 |
| 270° (3π/2) | undefined | undefined |
| 360° (2π) | 1 | 1 |
2√3/3 is the same number as 2/√3. Each secant is simply 1 divided by the cosine in the same row of a cosine table.
The Secant Graph
- Shape: U-shaped branches that open up where cos θ > 0 and down where cos θ < 0.
- Touches the cosine: sec θ and cos θ meet where cos θ = ±1, at 0°, 180° and 360°.
- Asymptotes: vertical lines at 90°, 270° and every 90° + k × 180°.
- Gap: no part of the graph lies between −1 and 1.
- Even function: sec(−θ) = sec θ, mirror-symmetric about the y-axis.
Secant Rules and Identities
- Reciprocal: sec θ = 1 ÷ cos θ.
- Pythagorean: 1 + tan2θ = sec2θ.
- Complement: sec(90° − θ) = csc θ.
- Negative angle: sec(−θ) = sec θ.
- Full turn: sec(θ + 360°) = sec θ.
- Calculus: the derivative of tan θ is sec2θ, and the derivative of sec θ is sec θ tan θ.
Worked Examples
Exact value of sec 30°.
= 1 ÷ (√3/2)
= 2/√3 = 2√3/3
≈ 1.1547
Length of a rafter. A roof slopes at 30° and covers 4 m horizontally. The sloping rafter is the hypotenuse:
= 4 × 1.1547
≈ 4.62 m
Radian inputs. sec(1) ≈ 1.8508, sec(2) ≈ −2.4030, sec(3) ≈ −1.0101, sec(4) ≈ −1.5299 and sec(5) ≈ 3.5253. 2, 3 and 4 radians lie between 90° and 270°, where cos θ is negative, so their secants are negative.
Secant in Excel and Code
| Tool | sec 60° |
|---|---|
| Excel / Google Sheets | =SEC(RADIANS(60)) |
| Older Excel | =1/COS(RADIANS(60)) |
| Python | 1 / math.cos(math.radians(60)) |
| JavaScript | 1 / Math.cos(60 * Math.PI / 180) |
| Scientific calculator | DEG mode, cos 60, then 1/x |
Python and JavaScript have no built-in secant, and all of these expect radians. Floating-point maths gives 1.9999999999999996 for sec 60° and about 1.63 × 1016 at 90° instead of an error.
Common Mistakes
- Confusing sec with cos⁻¹. sec θ = 1 ÷ cos θ; cos−1 (arccos) returns an angle.
- Mixing up secant and cosecant. Secant pairs with cosine, cosecant with sine.
- Expecting values below 1. A secant between −1 and 1 is always a mistake.
- Wrong angle mode. sec 60 is 2 in degrees but about −1.0500 in radians.
- Missing the undefined angles. sec 90° and sec 270° have no value.
Key Terms
- Hypotenuse
- The longest side of a right triangle, opposite the right angle.
- Adjacent
- The side next to θ that is not the hypotenuse.
- Reciprocal
- 1 divided by a value; secant is the reciprocal of cosine.
- Asymptote
- A line the graph approaches but never reaches; secant has them at 90° + k × 180°.
- Cosecant
- The reciprocal of sine, csc θ = 1 ÷ sin θ.
- Arcsec
- The inverse of secant, written sec⁻¹ or asec.
How to Use the Secant Calculator
- Enter the angle. Type the angle θ, for example 60, or press one of the angle buttons (30°, 45°, 60° and 90° in degrees, π/6, π/4, π/3 and π/2 in radians). In radians you can type a decimal such as 1 or a multiple of π such as pi/3 or 2π/3.
- Choose degrees or radians. Pick Degrees (°) or Radians (rad). Typing 60° or 1 rad also sets the unit for that angle.
- Read and copy the result. sec θ appears as you type, to 10 decimal places, with the exact value where there is one (for example sec(45°) = √2 ≈ 1.4142135624). The table below shows the angle in the other unit, the quadrant, cos θ and sin θ. At 90°, 270° and similar angles the secant is undefined and the calculator says so. Press Copy to copy the result, or Clear or Esc to start again.
Example: enter 120 in degrees and the result is −2, with the note that 120° is in quadrant II with a reference angle of 60°, and 120° = 2π/3 rad.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
What is the secant of an angle?
The secant is the reciprocal of the cosine: sec θ = 1 ÷ cos θ. In a right triangle it is the hypotenuse divided by the side next to the angle: sec θ = hypotenuse ÷ adjacent. For example sec 60° = 1 ÷ 0.5 = 2.
What is sec 0°, sec 30°, sec 45°, sec 60° and sec 90°?
sec 0° = 1, sec 30° = 2√3/3 ≈ 1.1547, sec 45° = √2 ≈ 1.4142 and sec 60° = 2. sec 90° is undefined, because cos 90° = 0. Also sec 180° = −1.
Why is sec 90° undefined?
sec θ = 1 ÷ cos θ and cos 90° = 0, so sec 90° would mean dividing by zero. Close to 90° the secant becomes huge (sec 89.9° ≈ 573), and the graph has vertical asymptotes at 90°, 270° and every 90° + k × 180°.
Can secant be between −1 and 1?
No. Cosine is always between −1 and 1, so its reciprocal is always 1 or more, or −1 or less: |sec θ| ≥ 1. The smallest positive value is sec 0° = 1 and the largest negative value is sec 180° = −1.
Is secant the same as cos⁻¹?
No. sec θ = 1 ÷ cos θ is the reciprocal of cosine. cos−1(x), or arccos, is the inverse function that returns an angle: cos−1(0.5) = 60°, while sec 60° = 2.
When is secant negative?
Secant has the same sign as cosine: positive in quadrants I and IV, negative in quadrants II and III. For example sec 120° = −2 and sec 135° = −√2 ≈ −1.4142.
What is the period of secant?
360°, or 2π radians, the same as cosine. sec(θ + 360°) = sec θ, so sec 420° = sec 60° = 2. Secant is also an even function: sec(−θ) = sec θ.
How do you calculate sec on a calculator or in Excel?
Most calculators have no sec key, so work out 1 ÷ cos θ. In Excel use =SEC(RADIANS(60)) or =1/COS(RADIANS(60)), both 2. In code use 1 / Math.cos(x) in JavaScript or 1 / math.cos(x) in Python, with x in radians; these give 1.9999999999999996 for 60° and a huge number instead of an error at 90°.
Where is the secant used?
The identity 1 + tan2θ = sec2θ is used to simplify expressions, and in calculus the derivative of tan θ is sec2θ. In geometry, sec θ gives the length of a slanted line for each unit of horizontal distance, such as a rafter: a roof at 30° needs 1.1547 m of rafter for every metre across.
Is the Secant Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Secant Calculator to find sec θ for any angle in degrees or radians instantly.