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Inverse Secant

Inverse Secant
sec-1
Sec-1
arcsec
Arcsec

The inverse function of secant.

Basic idea: To find sec-1 2, we ask "what angle has secant equal to 2?" The answer is 60°. As a result we say that sec-1 2 = 60°. In radians this is sec-1 2 = π/3.

More: There are actually many angles that have secant equal to 2. We are really asking "what is the simplest, most basic angle that has secant equal to 2?" As before, the answer is 60°. Thus sec-1 2 = 60° or sec-1 2 = π/3.

Details: What is sec-1 (–2)? Do we choose 120°, –120°, 240° , or some other angle? The answer is 120°. With inverse secant, we select the angle on the top half of the unit circle. Thus sec-1 (–2) = 120° or sec-1 (–2) = 2π/3.

In other words, the range of sec-1 is restricted to [0, 90°) U (90°, 180°] or The domain of inverse secant: [0, π/2) union (π/2, π]. Note: sec 90° is undefined, so 90° is not in the range of sec-1.

Note: arcsec refers to "arc secant", or the radian measure of the arc on a circle corresponding to a given value of secant.

Technical note: Since none of the six trig functions sine, cosine, tangent, cosecant, secant, and cotangent are one-to-one, their inverses are not functions. Each trig function can have its domain restricted, however, in order to make its inverse a function. Some mathematicians write these restricted trig functions and their inverses with an initial capital letter (e.g. Sec or Sec-1). However, most mathematicians do not follow this practice. This website does not distinguish between capitalized and uncapitalized trig functions.

 

Unit circle showing range of sec⁻¹: highlights 0 and π on horizontal axis, π/2 at top, excluding π/2 from range.

 

 

See also

Inverse trigonometry, inverse trig functions, interval notation

Key Formula

θ=sec⁡−1(x)⟺sec⁡(θ)=xDomain: x≤−1 or x≥1(i.e., ∣x∣≥1)Range: θ∈[0, π2)∪(π2, π]\begin{gathered}\theta = \sec^{-1}(x) \quad \Longleftrightarrow \quad \sec(\theta) = x\\\text{Domain: } x \le -1 \text{ or } x \ge 1 \quad (\text{i.e., } |x| \ge 1)\\\text{Range: } \theta \in \left[0,\,\tfrac{\pi}{2}\right) \cup \left(\tfrac{\pi}{2},\,\pi\right]\end{gathered}
Where:
  • θ\theta = The output angle, measured in radians (or degrees)
  • xx = The input value (the secant of the angle); must satisfy |x| ≥ 1

Worked Example

Problem: Find the exact value of sec⁻¹(2).
Step 1: Restate the problem: find the angle θ such that sec(θ) = 2.
sec⁡(θ)=2\sec(\theta) = 2
Step 2: Rewrite using cosine. Since sec(θ) = 1/cos(θ), we need cos(θ) = 1/2.
cos⁡(θ)=12\cos(\theta) = \frac{1}{2}
Step 3: Recall that cos(60°) = 1/2, so θ = 60°. Check the range: 60° lies in [0°, 90°), which is within the allowed range [0°, 90°) ∪ (90°, 180°].
θ=60°=π3\theta = 60° = \frac{\pi}{3}
Step 4: State the final result.
sec⁡−1(2)=π3\sec^{-1}(2) = \frac{\pi}{3}
Answer: sec⁻¹(2) = π/3 (or 60°)

Another Example

This example shows how to handle a negative input. With a negative argument, the angle falls in the second quadrant (between 90° and 180°), which is the key distinction from the first example.

Problem: Find the exact value of sec⁻¹(−2).
Step 1: Find the angle θ in the range [0, π/2) ∪ (π/2, π] such that sec(θ) = −2.
sec⁡(θ)=−2\sec(\theta) = -2
Step 2: Rewrite in terms of cosine: cos(θ) = −1/2.
cos⁡(θ)=−12\cos(\theta) = -\frac{1}{2}
Step 3: The cosine is negative, so θ must be in the second quadrant (the portion (90°, 180°] of the range). Recall that cos(120°) = −1/2.
θ=120°=2π3\theta = 120° = \frac{2\pi}{3}
Step 4: Verify: sec(120°) = 1/cos(120°) = 1/(−1/2) = −2. ✓
sec⁡−1(−2)=2π3\sec^{-1}(-2) = \frac{2\pi}{3}
Answer: sec⁻¹(−2) = 2π/3 (or 120°)

Frequently Asked Questions

What is the difference between inverse secant and inverse cosine?
Inverse secant and inverse cosine are closely related because sec(θ) = 1/cos(θ). You can convert between them using the identity sec⁻¹(x) = cos⁻¹(1/x). However, their domains differ: cos⁻¹ accepts inputs in [−1, 1], while sec⁻¹ accepts inputs where |x| ≥ 1. Their standard ranges also differ slightly—cos⁻¹ outputs [0, π], whereas sec⁻¹ outputs [0, π/2) ∪ (π/2, π], excluding π/2 because sec(π/2) is undefined.
Why is 90° (π/2) excluded from the range of inverse secant?
At 90°, cosine equals 0, which makes secant (1/cos) undefined. Since no real number has a secant that corresponds to 90°, this angle cannot be an output of sec⁻¹. The range therefore has a gap at π/2: [0, π/2) ∪ (π/2, π].
When do you use inverse secant?
Inverse secant appears in calculus, especially in certain integration formulas. A classic result is ∫ dx/(x√(x²−1)) = sec⁻¹|x| + C. It also arises in physics and engineering when solving equations that involve reciprocal trigonometric ratios.

Inverse Secant (sec⁻¹) vs. Inverse Cosine (cos⁻¹)

Inverse Secant (sec⁻¹)Inverse Cosine (cos⁻¹)
DefinitionAngle whose secant is xAngle whose cosine is x
Domain|x| ≥ 1 (i.e., x ≤ −1 or x ≥ 1)−1 ≤ x ≤ 1
Range[0, π/2) ∪ (π/2, π][0, π]
Key identitysec⁻¹(x) = cos⁻¹(1/x)cos⁻¹(x) = sec⁻¹(1/x)
Common in calculus∫ dx/(x√(x²−1)) = sec⁻¹|x| + Cd/dx [cos⁻¹(x)] = −1/√(1−x²)

Why It Matters

Inverse secant shows up frequently in integral calculus. The standard integral ∫ dx/(x√(x²−1)) has sec⁻¹|x| + C as its antiderivative, making this function essential for solving a family of integrals involving radicals. Beyond calculus, understanding inverse secant deepens your grasp of how all six trigonometric functions and their inverses are connected through reciprocal and complementary relationships.

Common Mistakes

Mistake: Confusing sec⁻¹(x) with 1/sec(x). Students sometimes read the −1 superscript as a reciprocal.
Correction: The notation sec⁻¹(x) means the inverse function (arcsec), not 1/sec(x). If you want the reciprocal, write [sec(x)]⁻¹ or simply cos(x).
Mistake: Choosing an angle outside the principal range, such as picking 240° for sec⁻¹(−2).
Correction: Always select the angle in [0°, 90°) ∪ (90°, 180°]. For negative inputs, the answer falls in the second quadrant (between 90° and 180°). So sec⁻¹(−2) = 120°, not 240°.

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