Sec Calculator

Free Online Sec Calculator — Solve sec(θ) Instantly

This sec calculator (secant calculator) is a complete trigonometry tool that solves secant problems in every direction you're likely to face — plug in an angle to get sec(θ), plug in the adjacent side and hypotenuse of a right triangle to get the angle, or plug in a decimal secant value to solve an equation like sec(θ) = 2 for θ. Every calculation is backed by a labeled diagram and a full step-by-step solution, so you can see exactly how the secant value or angle was derived instead of just getting a bare number.

Whether you're a student solving trigonometry homework, checking a physics or engineering problem, or just need a quick sin, cos, tan, and sec value for an angle in degrees or radians, this calculator covers the complete reciprocal-ratio picture in one place, with every step of the secant formula shown clearly.

What Is Secant (sec) in Trigonometry?

Secant is one of the three reciprocal trigonometric ratios, alongside cosecant and cotangent. For any angle θ measured in a right triangle, secant is defined as the ratio of the length of the hypotenuse to the length of the side adjacent to the angle — the exact reverse of the cosine ratio. This ratio, written sec(θ) = Hypotenuse ÷ Adjacent, is CAH flipped upside down. Notice that, unlike tangent and cotangent, secant absolutely needs the hypotenuse — it has no meaning without it:

• SOH — Sine = Opposite ÷ Hypotenuse
• CAH — Cosine = Adjacent ÷ Hypotenuse
• TOA — Tangent = Opposite ÷ Adjacent
• Secant = Hypotenuse ÷ Adjacent = 1 ÷ Cosine

The Secant Formula and the Unit Circle

Secant can also be written as the reciprocal of cosine: sec(θ) = 1 ÷ cos(θ). Geometrically, this has a striking origin story you can see live in the diagram above. Draw the vertical tangent line touching the unit circle at the point (1, 0). Now extend the ray from the origin through the angle θ until it crosses this tangent line — the length of the solid indigo segment from the origin to that crossing point is exactly sec(θ). This segment literally cuts across the inside of the circle, and that's precisely where the word 'secant' comes from — it derives from the Latin secare, meaning 'to cut'. Every value you enter updates this line live, so you can watch the segment stretch and shrink as sec(θ) changes.

• sec(θ) = Hypotenuse ÷ Adjacent (right-triangle definition)
• sec(θ) = 1 ÷ cos(θ) (reciprocal identity)
• sec(θ) = length of the segment from the origin to where the extended angle ray crosses the tangent line at x = 1
• θ = sec⁻¹(x) — the inverse secant (arcsec) function, used to solve for the angle when the ratio is known
• sec(θ) can never lie strictly between -1 and 1 — its range is (-∞, -1] ∪ [1, ∞)

How to Use This Sec Calculator — Step by Step

Pick the mode that matches your problem, enter your numbers, and read the result along with the diagram and worked solution below it.

• sec(θ) from an angle: type an angle and choose degrees, radians, or gradians — the calculator instantly returns sec(θ), cos(θ), sin(θ), the quadrant, and the reference angle, plotted on the secant-segment unit circle diagram.
• θ from right-triangle sides: enter the adjacent side and hypotenuse of a right triangle — the calculator applies sec(θ) = Hypotenuse ÷ Adjacent, then uses the inverse cosine to solve for θ, and draws the triangle to scale with every side labeled, including the computed opposite side.
• θ from a secant value: enter any real number with magnitude 1 or greater to solve equations like sec(θ) = 2 for θ, returning the principal angle plus the reflected solution on the other side of 360°.

Worked Example — Solving sec(60°)

Suppose you need sec(60°). Switch to the first mode, enter 60, choose degrees, and the calculator converts 60° to π/3 radians, finds cos(60°) = 0.5, and takes the reciprocal: sec(60°) = 1 ÷ 0.5 = 2 exactly. In the diagram, the solid segment from the origin stretches out to meet the tangent line precisely at a length of 2 circle-radii, matching this well-known result.

Now suppose you're given a right triangle with an adjacent side of 4 and a hypotenuse of 5 — a classic 3-4-5 triangle. Switching to the second mode and entering those values gives sec(θ) = 5 ÷ 4 = 1.25, and applying the inverse cosine, θ = cos⁻¹(4 ÷ 5) = cos⁻¹(0.8) ≈ 36.87°. The calculator also finds the opposite side using the Pythagorean theorem: √(5² - 4²) = 3, completing the full 3-4-5 right triangle so every side and angle is visible on the diagram.

Common Secant Values Table

These standard angles come up constantly in trigonometry, and it helps to recognize them at a glance rather than recalculating them each time:

• sec(0°) = 1
• sec(30°) = 2/√3 ≈ 1.155
• sec(45°) = √2 ≈ 1.414
• sec(60°) = 2
• sec(90°) = undefined
• sec(180°) = -1
• sec(270°) = undefined
• sec(360°) = 1

Why Is sec(90°) Undefined?

Since sec(θ) = 1 ÷ cos(θ), secant becomes undefined whenever cos(θ) = 0 — which happens at 90°, 270°, and every odd multiple of 90° after that, exactly the same points where tangent is undefined. Geometrically, at exactly 90° the angle's ray points straight up, running parallel to the tangent line rather than crossing it, so there's no intersection point and no finite secant length. On a graph of sec(θ), these points appear as vertical asymptotes, where the curve rockets toward positive or negative infinity. This calculator detects these angles automatically and reports 'undefined' instead of a misleading number.

Understanding the Sign and Range of sec(θ)

Because sec(θ) = 1 ÷ cos(θ), its sign always matches the sign of cos(θ). This calculator automatically detects and displays the quadrant for any angle you enter:

• Quadrant 1 (0°-90°): cos is positive, so sec(θ) is positive
• Quadrant 2 (90°-180°): cos is negative, so sec(θ) is negative
• Quadrant 3 (180°-270°): cos is negative, so sec(θ) is negative
• Quadrant 4 (270°-360°): cos is positive, so sec(θ) is positive

The Forbidden Gap Between -1 and 1

Because cosine is always bounded between -1 and 1, its reciprocal, secant, can never fall strictly between -1 and 1 — sec(θ) always satisfies sec(θ) ≤ -1 or sec(θ) ≥ 1. This calculator's third mode enforces this rule automatically and flags any entered value inside that forbidden gap as having no real solution.

Secant vs Cosine — How They Relate

Secant and cosine are reciprocals of each other: sec(θ) = 1 ÷ cos(θ), and cos(θ) = 1 ÷ sec(θ). This means whenever cos(θ) is close to 1, sec(θ) is also close to 1, and whenever cos(θ) shrinks toward zero, sec(θ) grows without bound. Both functions share the same period of 360°, and sec(θ) is undefined at exactly the angles where cos(θ) = 0.

Degrees vs Radians vs Gradians

Angles can be measured in more than one unit, and this calculator supports the three most common ones. Degrees split a full circle into 360 equal parts and are the most familiar unit in everyday geometry. Radians measure angle by arc length relative to the radius, making a full circle equal to 2π radians — this is the unit calculus and most programming languages use internally. Gradians (or gons) split a full circle into 400 parts and appear mostly in surveying and some European engineering contexts. Whichever unit your problem uses, simply select it from the dropdown and the calculator handles the conversion automatically.

Where Secant Calculations Are Used in Real Life

Secant isn't just a classroom exercise — it shows up anywhere a reciprocal distance or scaling factor matters:

• Optics and lens design: secant relationships describe how light bends and focuses through curved surfaces at an angle.
• Surveying and mapping: secant projections (like the secant cone or secant cylinder) are used to minimize distortion when projecting the curved Earth onto a flat map.
• Physics and engineering: secant appears in problems involving oblique forces, inclined planes, and structural load angles.
• Calculus: secant is a standard substitution function in integration problems, especially those involving √(x² - a²).
• Navigation and astronomy: secant-based formulas historically helped compute distances and positions from angular measurements.

Tips for Getting Accurate Results

Always double-check which angle unit your problem is stated in before entering a value — a common mistake is entering a radian value while the unit dropdown is still set to degrees, which will give a wildly different (and wrong) answer. When solving for an angle from a secant value, remember that sec(θ) can never be strictly between -1 and 1, so if your equation seems to require that, double-check your numbers — there's no real angle that satisfies it. Also remember that sec(θ) is simply 1 ÷ cos(θ), so if you already know an angle's cosine, you can find its secant instantly by taking the reciprocal — no calculator required for that last step.