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Secant

secant
sec

The trig function secant, written sec θ. sec θ equals The fraction 1 over cos θ, representing the formula for secant: sec θ = 1/cos θ. For acute angles, sec θ can be found by the SOHCAHTOA definition as shown below on the left. The circle definition, a generalization of SOHCAHTOA, is shown below on the right. f(x) = sec x is a periodic function with period 2π.

Two diagrams defining sec θ: a right triangle with hypotenuse, opposite, adjacent sides (sec θ = hypotenuse/adjacent); a unit...

 

 

See also

Inverse secant, angle

Key Formula

sec⁡θ=1cos⁡θ=hypotenuseadjacent\sec\theta = \frac{1}{\cos\theta} = \frac{\text{hypotenuse}}{\text{adjacent}}
Where:
  • θ\theta = The angle in a right triangle or on the unit circle
  • hypotenuse\text{hypotenuse} = The longest side of the right triangle, opposite the right angle
  • adjacent\text{adjacent} = The side of the right triangle next to the angle θ (not the hypotenuse)

Worked Example

Problem: In a right triangle, the side adjacent to angle θ has length 3 and the hypotenuse has length 5. Find sec θ.
Step 1: Recall the right-triangle definition of secant: hypotenuse over adjacent.
sec⁡θ=hypotenuseadjacent\sec\theta = \frac{\text{hypotenuse}}{\text{adjacent}}
Step 2: Substitute the given side lengths into the formula.
sec⁡θ=53\sec\theta = \frac{5}{3}
Step 3: Verify using the reciprocal relationship. First find cosine, then take its reciprocal.
cos⁡θ=35⇒sec⁡θ=135=53\cos\theta = \frac{3}{5} \quad\Rightarrow\quad \sec\theta = \frac{1}{\frac{3}{5}} = \frac{5}{3}
Answer: sec θ = 5/3

Another Example

This example uses an obtuse angle (beyond 90°) to show how the unit-circle definition extends secant past acute angles, and how reference angles and quadrant signs determine the result.

Problem: Evaluate sec(120°) exactly.
Step 1: Since 120° is not an acute angle, use the unit-circle definition: sec θ = 1/cos θ.
sec⁡(120°)=1cos⁡(120°)\sec(120°) = \frac{1}{\cos(120°)}
Step 2: Find cos(120°). The reference angle is 60°, and 120° lies in Quadrant II where cosine is negative.
cos⁡(120°)=−cos⁡(60°)=−12\cos(120°) = -\cos(60°) = -\frac{1}{2}
Step 3: Take the reciprocal to get secant.
sec⁡(120°)=1−12=−2\sec(120°) = \frac{1}{-\frac{1}{2}} = -2
Answer: sec(120°) = −2

Frequently Asked Questions

What is the difference between secant and cosecant?
Secant is the reciprocal of cosine: sec θ = 1/cos θ. Cosecant is the reciprocal of sine: csc θ = 1/sin θ. They are reciprocals of different base functions, so they have different values, different domains, and different graphs. A common memory trick: the 'co-' in cosecant pairs it with sine (its complement), while secant pairs with cosine.
When is secant undefined?
Secant is undefined wherever cosine equals zero, because you cannot divide by zero. This happens at θ = 90° + 180°·n (or in radians, θ = π/2 + nπ) for any integer n. On the graph of sec x, these points correspond to vertical asymptotes.
Why is the range of secant (−∞, −1] ∪ [1, ∞)?
Because cosine ranges from −1 to 1, its reciprocal can never produce a value between −1 and 1. When cos θ is close to 0, sec θ grows very large in magnitude. When |cos θ| = 1, sec θ = ±1. So sec θ is always ≤ −1 or ≥ 1.

Secant (sec θ) vs. Cosine (cos θ)

Secant (sec θ)Cosine (cos θ)
Definitionsec θ = 1/cos θ = hyp/adjcos θ = adj/hyp
Range(−∞, −1] ∪ [1, ∞)[−1, 1]
Period2π2π
Undefined atθ = π/2 + nπ (where cos θ = 0)Defined for all real θ
Graph shapeU-shaped curves with vertical asymptotesSmooth continuous wave

Why It Matters

Secant appears frequently in calculus, especially in integrals and derivatives involving trigonometric substitution. The derivative of tan x is sec² x, making secant essential for computing areas under curves and solving differential equations. You will also encounter secant in physics when analyzing forces at angles and in engineering when working with waveforms and signal processing.

Common Mistakes

Mistake: Confusing sec θ with csc θ — thinking secant is the reciprocal of sine.
Correction: Secant is the reciprocal of cosine (sec = 1/cos). Cosecant is the reciprocal of sine (csc = 1/sin). Remember: the function without the 'co-' prefix is the reciprocal of the other function without 'co-'. Secant goes with cosine; cosecant goes with sine.
Mistake: Writing sec θ = cos⁻¹ θ and confusing it with the inverse cosine (arccos).
Correction: sec θ = (cos θ)⁻¹ = 1/cos θ, which is a reciprocal. The notation cos⁻¹ θ typically means arccos θ (the inverse function that returns an angle). These are completely different operations. To avoid ambiguity, use the fraction 1/cos θ for secant.

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