The secant function ‘or’ **Sec Theta** is one of the trigonometric functions apart from sine, cosine, tangent, cosecant, and cotangent. In right-angled trigonometry, the secant function is defined as the ratio of the hypotenuse and adjacent side.

The mathematical denotation of the secant function is,

\(\sec(\theta) = \frac{\text{Hypotenuse Side}}{\text{Adjacent Side}}\)Index

## More About Sec Theta

The derivative of \(\sec(\theta)\) in calculus is \(\sec(\theta) \tan(\theta)\) and the integral of it is \(\ln|\sec(\theta) + \tan(\theta)|\). The reciprocal of \(\sec(\theta)\) is \(\cos(\theta)\).

Below is a table of sec theta values for different degrees and radians.

Radians | Degree | Tangent Value |

0 | 0° | 1 |

\(\frac{\pi}{6}\) | 30° | \(\frac{2}{\sqrt{3}}\) |

\(\frac{\pi}{4}\) | 45° | \(\sqrt{2}\) |

\(\frac{\pi}{3}\) | 60° | 2 |

\(\frac{\pi}{2}\) | 90° | \(\infty\) |

\(\pi\) | 180° | -1 |

\(\frac{3\pi}{2}\) | 270° | \(\infty\) |

\(2\pi\) | 360° | 1 |

## Important Sec Theta Formula

Some important properties of the secant function and sec theta formula are:

- \(\sec(-x) = \sec(x)\)

- \(\sec(90° – x) = cosec(x)\)

- \(\sec(x + 2\pi) = \sec(x)\)

- \(\sec(\pi – x) = -\sec(x)\)

- \(\sec^2(x) = 1 + \tan^2(x)\)

- \(sec(x + y) = \frac{\sec(x) \sec(y)}{1 – \tan(x) \tan(y)}\)

**Solved Examples**

**Question 1. If \(\cos(x) = \frac{3}{5}\), calculate the value of \(\sec(x)\).**

**Solution.** Using trigonometric identity,

\(\sec(x) = \frac{1}{\cos(x)} = \frac{1}{3/5}\)

\(∴ \sec(x) = \frac{5}{3}\)

**Question 2. If \(\tan(\frac{x}{2}) = \frac{5}{8}\), calculate the value of \(\sec(x)\).**

**Solution.** Using trigonometric identity,

\begin{align}

\sec^2(x) & = 1 + \tan^2(x)\\

& = 1 + (\frac{5}{8})^2\\

& = \frac{89}{64}\\

\sec(x) & = \frac{\sqrt{89}}{8}\\

\end{align}

\)

**Question 3. Prove \(\sec(x + y) = \frac{\sec(x) \sec(y)}{1 – \tan(x) \tan(y)}\)**

**Solution.** As we know,

\(\sec(x) = \frac{1}{\cos(x)}\) and \(\tan(x) = \frac{\sin(x)}{\cos(x)}\)

Considering RHS of given equation,

\(\begin{align}

\frac{\sec(x) \sec(y)}{1 – \tan(x) \tan(y)} & = \frac{\frac{1}{\cos(x)} \times \frac{1}{\cos(y)}}{1 -\frac{\sin(x)}{\cos(x)} \times \frac{\sin(x)}{\cos(y)}}\\

& = \frac{1}{\cos(x) \cos(y) – \sin(x) \sin(y)}\\

& = \frac{1}{\cos(x + y)}\\

& = \sec(x + y)\\

\end{align}

\)

Hence, LHS = RHS … **Proved**

**FAQs**

**Explain how sec(-x) = sec(x****).**As we know, the angle (-x) lies in the 4th quadrant of a graph, and the secant is positive in this quadrant. Hence, sec(-x) = sec(x).

**What is sec theta****?**Sec theta of an angle in a right-angled triangle is defined as the ratio of the hypotenuse and adjacent side.

**In which quadrants is the secant function positive and in which quadrants is it negative?**

It can be observed from the above graph that sec(x) is positive in the 1st and 4th quadrants and negative in the 2nd and 3rd quadrants.

**Know More Trigonometric Function**