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Sec Theta

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.

Right angled triangle -Sec Theta Definition

The mathematical denotation of the secant function is,

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

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 \(\cos(\theta)\).

Graph of the sec theta function
Graph of the sec theta function

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

RadiansDegreeTangent Value
01
\(\frac{\pi}{6}\)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?

Tan-function-sign-quadrants
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

Sin ThetaCos Theta
Tan ThetaCot Theta
Cosec Theta
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