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The Secant Calculator evaluates the secant of any angle in degrees or radians. Secant (abbreviated sec) is the reciprocal of cosine and appears extensively in calculus, differential equations, and advanced geometry.
Secant is defined as:
$$\sec\theta = \frac{1}{\cos\theta} = \frac{\text{hypotenuse}}{\text{adjacent}}$$
Since secant is the reciprocal of cosine, it is undefined wherever cosine equals zero — at $$\theta = 90°$$ and $$\theta = 270°$$ (and their periodic equivalents $$90° + n \cdot 180°$$). At these angles, secant has vertical asymptotes.
The range of secant mirrors that of cosecant: $$(-\infty, -1] \cup [1, +\infty)$$. The absolute value is always at least 1. Secant equals exactly 1 at $$\theta = 0°$$ and $$-1$$ at $$\theta = 180°$$.
Key exact values include:
$$\sec 0° = 1, \quad \sec 30° = \frac{2}{\sqrt{3}}, \quad \sec 45° = \sqrt{2}, \quad \sec 60° = 2, \quad \sec 90° = \text{undefined}$$
Like cosine, secant is an even function: $$\sec(-\theta) = \sec\theta$$. It has a period of $$2\pi$$ radians (360°). Secant is positive in Quadrants I and IV (where cosine is positive) and negative in Quadrants II and III.
Secant satisfies the Pythagorean identity:
$$\tan^2\theta + 1 = \sec^2\theta$$
This identity, derived by dividing $$\sin^2\theta + \cos^2\theta = 1$$ by $$\cos^2\theta$$, is one of the most frequently used results in calculus. It directly gives the derivative of tangent: $$\frac{d}{dx}\tan(x) = \sec^2(x)$$.
In calculus, the secant function's own derivative is $$\frac{d}{dx}\sec(x) = \sec(x)\tan(x)$$, and its integral is $$\ln|\sec(x) + \tan(x)| + C$$. The latter integral is notoriously tricky to derive and is a classic exercise in integration techniques. Secant also appears prominently in trigonometric substitution, where $$x = a\sec\theta$$ is used to evaluate integrals involving $$\sqrt{x^2 - a^2}$$.
In physics, secant appears in the secant method for root-finding (named for the geometric secant line, not the trig function directly), in optics formulas for refraction angles, and in projections used for map-making. The Mercator projection, used in navigation, involves the integral of secant, which equals $$\ln|\sec\theta + \tan\theta|$$.
This calculator displays both the secant and cosine values, making it easy to verify the reciprocal relationship $$\sec\theta \times \cos\theta = 1$$.
The calculator converts the angle to radians if necessary, evaluates $$\cos(\theta)$$, then computes $$\sec\theta = \frac{1}{\cos\theta}$$. If $$|\cos\theta| < 10^{-10}$$, the result is undefined to avoid numerical instability near vertical asymptotes.
Secant is always $$\geq 1$$ or $$\leq -1$$, never between. An undefined result means cosine is zero at that angle (90°, 270°). Values close to 1 or −1 indicate that cosine is near its extremes. Very large values indicate cosine is near zero.
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sec(60°) = 1/cos(60°) = 1/0.5 = 2. This is a standard exact value. Note that sec(60°) = csc(30°) = 2 by the co-function relationship.
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In Quadrant II, cosine is negative, so secant is also negative. sec(135°) = 1/cos(135°) = 1/(−√2/2) = −√2 ≈ −1.4142.
Since $$\sec\theta = \frac{1}{\cos\theta}$$, it is undefined where $$\cos\theta = 0$$, which occurs at $$90°$$ and $$270°$$ (plus multiples of 360°). At these points, secant approaches $$\pm\infty$$ and the graph has vertical asymptotes.
Secant is an even function because $$\sec(-\theta) = \frac{1}{\cos(-\theta)} = \frac{1}{\cos\theta} = \sec\theta$$. It inherits this property from cosine, which is also even. The secant graph is symmetric about the y-axis.
The Pythagorean identity $$\tan^2\theta + 1 = \sec^2\theta$$ connects secant and tangent. This means $$\sec\theta = \pm\sqrt{1 + \tan^2\theta}$$, with the sign determined by the quadrant. This identity is essential in calculus for finding the derivative of tangent.
Secant is critical in calculus: $$\frac{d}{dx}\tan(x) = \sec^2(x)$$, and the trigonometric substitution $$x = a\sec\theta$$ handles integrals with $$\sqrt{x^2 - a^2}$$. The integral $$\int \sec(x)\,dx = \ln|\sec(x) + \tan(x)| + C$$ appears in arc length calculations and the Mercator map projection.
The secant graph consists of U-shaped curves alternating above 1 and below −1, separated by vertical asymptotes at every odd multiple of 90°. Between −90° and 90°, the curve opens upward with minimum value 1 at 0°. Between 90° and 270°, it opens downward with maximum value −1 at 180°.
Secant and cosecant are co-functions: $$\sec\theta = \csc(90° - \theta)$$. This means the secant of an angle equals the cosecant of its complement. For example, $$\sec 60° = \csc 30° = 2$$.
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