Math
Secant, Cosecant, and Cotangent Calculator
Secant, cosecant, and cotangent are the reciprocals of cosine, sine, and tangent.
Enter any angle to get all three with exact, rationalized forms for special angles (sec 30° = 2√3/3, csc 45° = √2, cot 60° = √3/3), clear undefined flags where the denominator is zero, and the primary values they come from.
Type a number, a fraction like 180/7, a π expression like 5π/6, or degrees-minutes-seconds like 30° 15′ 30″. Negative and large angles are fine.
Try an example
Result
sec θ
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- csc θ
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- cot θ
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- sin θ
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- cos θ
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- tan θ
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- Quadrant
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More details (1 more)
- Radians
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Student quick launch
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Study path
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Definitions
Each The number you get by flipping a fraction; 3/4 becomes 4/3, and 5 becomes 1/5. function is undefined exactly where its partner is zero: sec at 90° and 270°, csc and cot at 0° and 180°. Because |sin θ| and |cos θ| never exceed 1, |sec θ| and |csc θ| are never less than 1.
Exact values
| θ | sec θ | csc θ | cot θ |
|---|---|---|---|
| 0° | 1 | undefined | undefined |
| 30° | 2√3/3 | 2 | √3 |
| 45° | √2 | √2 | 1 |
| 60° | 2 | 2√3/3 | √3/3 |
| 90° | undefined | 1 | 0 |
| 120° | -2 | 2√3/3 | -√3/3 |
| 135° | -√2 | √2 | -1 |
| 150° | -2√3/3 | 2 | -√3 |
| 180° | -1 | undefined | undefined |
How to use it
- Enter the angle in degrees or An angle unit where a full turn is 2π instead of 360 degrees..
- Read sec, csc, and cot; each shows the The answer kept as a fraction or radical instead of a rounded decimal. and a decimal.
- Open Show the work to see the reciprocal taken and, for special angles, the rationalization step.
How to read the answer
- sec θ has the same sign as cos θ; csc θ the same sign as sin θ; cot θ the same sign as tan θ.
- Rationalized forms: 1/(√3/2) = 2/√3 = 2√3/3. Both are correct, but the rationalized form is the one teachers expect.
- undefined is not a huge number; it means the function has a vertical asymptote at that angle.
Common mistakes and edge cases
- Confusing sec with the reciprocal of sine: sec pairs with cos, csc pairs with sin (the letters do not match).
- Writing sin⁻¹ θ for csc θ: sin⁻¹ is arcsine, an angle, not a reciprocal.
- Reporting cot 90° as undefined: cot 90° = cos 90° / sin 90° = 0/1 = 0. It is tan 90° that is undefined.
- Forgetting to rationalize the The bottom number of a fraction; it says how many equal parts make one whole. in exact answers.
Worked examples
sec 30°
sec θ
2√3/3 (1.154701)
sec 45°
sec θ
√2 (1.414214)
Quadrant II
sec 120° = -2
sec θ
-2
Radians
sec(5π/6) = -2√3/3
sec θ
-2√3/3 (-1.154701)
Undefined secant
sec 90°
sec θ
undefined
Undefined cosecant
θ = 0°: sec is 1 but csc and cot are undefined
sec θ
1
Decimal angle
sec 40°
sec θ
1.305407
Unreadable input
sec θ
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Frequently asked questions
What is secant?+
The reciprocal of cosine: sec θ = 1 / cos θ. In a right triangle it is hypotenuse divided by adjacent.
What is sec 30 degrees?+
sec 30° = 1 / cos 30° = 1 / (√3/2) = 2/√3 = 2√3/3 ≈ 1.1547.
When is cosecant undefined?+
Whenever sin θ = 0: at 0°, 180°, 360°, and every multiple of 180°. Cotangent is undefined at the same angles.
Is cot θ = 1 / tan θ always true?+
Wherever both are defined. At 90°, tan θ is undefined but cot θ = 0, so the safer definition is cot θ = cos θ / sin θ.
Can sec θ be between -1 and 1?+
No. Since |cos θ| ≤ 1, |sec θ| ≥ 1. The same holds for cosecant. Cotangent, however, can take any real value.
How do I rationalize 1/(√2/2)?+
1/(√2/2) = 2/√2. Multiply top and bottom by √2: 2√2/2 = √2. So sec 45° = csc 45° = √2.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026