Secant Calculator

Definition and Formula of Secant

The secant function (\(\sec(\theta)\)) is one of the fundamental trigonometric functions. Here, \(\theta\) is the angle, typically measured in radians.

In a right triangle, the secant function is defined as the ratio of the hypotenuse to the adjacent side: \( \sec(\theta) = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{c}{b} \) This represents the ratio of the hypotenuse to the adjacent side of the given angle.

In the unit circle, the secant function is the reciprocal of the cosine function: \( \sec(\theta) = \frac{1}{\cos(\theta)} \)

Examples

Example 1: Secant Value in a Right Triangle

Find the secant value for an angle \(\theta = 60^\circ\) in a right triangle where the adjacent side measures 2 units and the hypotenuse is 4 units.

Example 2: Real-Life Application

Calculate the secant value for a slope with an angle of \(\theta = 30^\circ\).

Secant Graph and Properties

Key Characteristics:

• Periodicity: The secant function has a period of \(2\pi\) (360°), meaning it repeats every \(2\pi\) radians.
• Even Function: The secant function is symmetric about the y-axis, satisfying \(\sec(-\theta) = \sec(\theta)\).
• Amplitude: The secant function's values range from negative infinity to positive infinity.
• Vertical Asymptotes: At \(\theta = \frac{\pi}{2} + n\pi\) (where \(n\) is an integer), secant has vertical asymptotes as \(\cos(\theta) = 0\).
• Domain and Range
  • Domain: All angles except \(\theta = \frac{\pi}{2} + n\pi\) (\(n \in \mathbb{Z}\)).
  • Range: \((-\infty, -1] \cup [1, \infty)\).

Secant in Quadrants

Other Secant Calculations

1. Reciprocal of Secant (Cosine Function)

The reciprocal of the secant function is the cosine function: \( \frac{1}{\sec(\theta)} = \cos(\theta) \) Note: When \(\sec(\theta) = 0\), the cosine function is undefined.

2. Derivative of Secant

The derivative of the secant function is: \( \frac{d}{d\theta} \sec(\theta) = \sec(\theta) \tan(\theta) \) This property is useful in calculus for analyzing rates of change.

3. Integral of Secant

The integral of the secant function is: \( \int \sec(\theta) \, d\theta = \ln|\sec(\theta) + \tan(\theta)| + C \)

4. Inverse Secant (Arcsec)

The inverse secant function (\(\text{arcsec}(x)\)) calculates the angle for a given secant value: \( \theta = \text{arcsec}(x) \)

Common Secant Values Table