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Zuletzt aktualisiert: 19. August 2026

Sekans Rechner

Quick Answer

Secant is the reciprocal of cosine: sec(θ) = 1/cos(θ). This calculator evaluates secant for degree inputs, shows the cosine value used in the reciprocal, and rejects undefined angles where cosine is zero.

The Secant Calculator finds secant by taking one divided by cosine, and it flags angles like 90 degrees where the function is undefined.

Wichtige Erkenntnisse

  • Secant is defined by sec(θ) = 1/cos(θ).
  • The sign of secant follows the sign of cosine.
  • Secant is undefined wherever cosine equals zero.
  • Reference angles help predict exact values and signs by hand.
  • The coterminal-angle output makes periodicity easier to recognize.
Dharmendra Singh - Creator
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Formel

sec(θ) = 1 / cos(θ)

Wobei:

  • θ=Angle in degrees(degrees)
  • sec(θ)=Secant of the angle
  • cos(θ)=Cosine of the angle
Secant from CosineA right-style angle diagram highlights an angle theta. The adjacent side is labeled cosine, and a separate formula card states that secant is one divided by cosine. A warning card reminds the reader that secant is undefined when cosine equals zero.Secant in Degrees — Reciprocal of Cosineθadjacent → cos(θ)radius / hypotenuseCore identitysec(θ)=1/cos(θ)Undefined whencos(θ)=0
Secant is not a separate mystery function: compute cosine first, then take its reciprocal. The key restriction is that angles making cosine equal zero also make secant undefined.

Rechenbeispiele

Secant of 60°

A classic special-angle example where cosine is positive and easy to verify.

  1. 1Use the identity sec(θ) = 1/cos(θ).
  2. 2cos(60°) = 1/2.
  3. 3Take the reciprocal: 1 ÷ 1/2 = 2.
Endergebnis: sec(60°) = 2

Secant of 120°

Use a second-quadrant angle to see the sign change.

  1. 1cos(120°) = -1/2 because the reference angle is 60° in quadrant II.
  2. 2Secant is the reciprocal of cosine.
  3. 31 ÷ (-1/2) = -2, so sec(120°) = -2.
Endergebnis: sec(120°) = -2

Secant of 300°

A fourth-quadrant example with the same reference angle as 60°.

  1. 1The reference angle is 60° and cosine is positive in quadrant IV.
  2. 2cos(300°) = 1/2.
  3. 3Take the reciprocal to get sec(300°) = 2.
Endergebnis: sec(300°) = 2

Einführung

The Secant Calculator evaluates the trigonometric secant function for angles measured in degrees. Secant is defined by the reciprocal identity sec(θ) = 1/cos(θ), so the entire problem really comes down to understanding cosine and then checking whether cosine is zero. That domain check matters, because secant is undefined whenever cosine vanishes, such as at 90°, 270°, and all coterminal angles that differ from those values by full turns. This page therefore does more than display a number: it shows the cosine used to build the answer, returns a coterminal angle to make periodicity visible, and explains the sign behavior that changes across quadrants.

What the secant function measures

Secant is one of the reciprocal trigonometric functions. If cosine tells you a ratio or coordinate component, secant tells you its reciprocal. In practice, secant is often easier to understand when you remember that it is built from cosine, not defined independently. That viewpoint makes the domain restrictions and sign changes much easier to predict.

  • sec(θ) is the reciprocal of cos(θ).

  • If cosine is positive, secant is positive.

  • If cosine is negative, secant is negative.

  • If cosine is zero, secant is undefined.

Understanding sec(θ) = 1/cos(θ)

The formula sec(θ) = 1/cos(θ) says two important things at once. First, secant inherits its period and quadrant signs from cosine. Second, secant becomes very large in magnitude when cosine gets very close to zero, because reciprocals of small numbers grow quickly. That is why values near 90° and 270° can change dramatically even when the angle moves only a little.

  • Periodicity follows cosine.

  • Small cosine magnitude means large secant magnitude.

  • Exact special-angle values are often easy to verify by hand.

  • Undefined points occur at odd multiples of 90°.

Why this calculator uses degrees

Degrees are the most common classroom angle unit for hand-checked trigonometry examples, so this calculator accepts degree input directly. That keeps the examples readable and makes special angles like 30°, 45°, 60°, 120°, and 300° easy to recognize. If you begin from a radian problem, convert the angle first or use the periodicity idea to find an equivalent degree measure for quick checking.

  • 60° is a standard benchmark angle.

  • 120° and 300° share the 60° reference angle.

  • 90° signals an undefined secant value.

  • Coterminal angles differ by multiples of 360°.

Worked example: sec(120°)

Start with the identity sec(θ) = 1/cos(θ). The angle 120° lies in the second quadrant, where cosine is negative. Its reference angle is 60°, and cos(60°) = 1/2, so cos(120°) = -1/2. Taking the reciprocal gives sec(120°) = -2. This is a good example because it shows that the magnitude comes from the reference angle while the sign comes from the quadrant.

  • Reference angle: 60°.

  • Quadrant: II.

  • cos(120°) = -1/2.

  • sec(120°) = -2.

When secant is undefined

Secant is undefined whenever cosine equals zero. In degree measure, that happens at 90°, 270°, 450°, and every other angle coterminal with those benchmarks. Since dividing by zero is impossible in ordinary real-number arithmetic, the calculator rejects those inputs instead of pretending that a huge number is a valid answer. That guard rule is essential when graphs or numerical approximations make a near-undefined angle look harmless.

  • 90° is undefined.

  • 270° is undefined.

  • Adding 360° keeps the same undefined status.

  • Near-undefined is not the same as defined.

Common mistakes

Students often confuse secant with cosecant because both are reciprocal trig functions. Another mistake is using the reference angle correctly but forgetting to apply the quadrant sign. A third is reading a calculator output near 90° as if it were exact, even though the true function is undefined there. Keeping the cosine output visible helps catch all three errors quickly.

  • Do not mix up secant and cosecant.

  • Do not ignore the quadrant sign.

  • Do not trust a near-vertical reciprocal value without a domain check.

  • Do not forget that coterminal angles repeat every 360°.

When people use secant values

Secant appears in trigonometry, analytic geometry, calculus, and applications involving reciprocal cosine relationships. Even when a later formula hides secant inside algebraic manipulation, recognizing the reciprocal structure helps with simplification and graph interpretation. This page is most useful when you need a quick degree-based check before moving on to an identity, graph, or applied problem.

  • Trig homework and quizzes.

  • Identity verification.

  • Graph interpretation near asymptotes.

  • Checking algebraic substitutions that involve reciprocal cosine.

Hand method versus calculator method

For special angles, solving by hand is still the best way to learn secant. Angles such as 60°, 120°, and 300° have cosine values that are easy to recall, so the reciprocal can be checked mentally. The calculator becomes more valuable when the angle is not special, when you want the coterminal-angle reminder, or when you need to verify that the input is not sitting on an undefined point.

  • Use hand work for standard angles.

  • Use the calculator for non-special decimals in degrees.

  • Always inspect the cosine output before trusting the reciprocal.

  • Use the coterminal angle to recognize repeating patterns.

Kurzreferenzkarte

Secant quick reference

Kurzreferenz • Sekans Rechner

sec(θ) = 1 / cos(θ)

Gültiger Bereich: Use any finite angle in degrees except angles coterminal with 90° or 270°.

Häufige Werte

sec(60°)2
sec(120°)-2
sec(300°)2
Default inputsec(60°) = 2

⚠ Achtung

  • •Angles where cosine is zero make secant undefined.
  • •Do not confuse secant with cosecant.
  • •Near 90° and 270°, tiny angle changes can produce huge secant changes.
  • •Reference-angle magnitude is not enough; quadrant sign still matters.

Profi-Tipps

  • →Find cosine first, then take the reciprocal.
  • →Use coterminal angles to simplify large or negative degree inputs.
  • →Memorize secant values for common special angles by using known cosine values.
  • →Check whether the answer should be positive or negative before calculating the exact value.

FAQ

What is secant in trigonometry?

Secant is the reciprocal of cosine, defined by sec(θ) = 1/cos(θ).

Why is secant undefined at 90°?

Because cos(90°) = 0, and dividing 1 by 0 is undefined in real-number arithmetic.

Does secant have the same period as cosine?

Yes. Since secant is built directly from cosine, its values repeat every 360° in degree measure.

Why can secant be negative?

Secant takes the sign of the reciprocal of cosine, so it is negative in quadrants where cosine is negative.

How can I estimate a secant value quickly?

Estimate cosine first using a reference angle and quadrant, then take its reciprocal.

Can I use coterminal angles with this calculator?

Yes. Coterminal angles have the same secant value because they have the same cosine value.

What is the difference between secant and cosecant?

Secant is the reciprocal of cosine, while cosecant is the reciprocal of sine.

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