The secant (sec) of an angle in a right triangle is defined as the ratio between the length of the hypotenuse and the length of the adjacent side. To put it in simpler terms, secant is the inverse of the cosine function.
Let's consider a right-angled triangle ABC with the right angle at B and an acute angle (θ) at B as depicted in the figure below.
Sec θ = Hypotenuse / Adjacent side
We can replace Secant with cos x in the following manner, Sec θ = 1 / Cos θ.
Sec 90 Degrees
The value of sec 90 degrees is undefined.
Sec 90 Value
The value of Sec 90 (in radians) = -2.2317761286…
The Inverse Sec x Formula
For every trigonometric function, there exists an inverse function that works in reverse. The inverse function of sec is known as arcsec.
The value of Secant 90 degree cannot be calculated and is undefined in the trigonometric table.
Calculating Sec 90
Since the angle lies between 0 and 90 degrees, it is located in the 1st quadrant, where the values of sin, cos, and tan are positive.
90 degrees is a right angle.
Sec x = 1/cos x
Sec 90° = 1/ cos 90°
Sec 90° = 1/ 0
Sec 90° = undefined
Sec 90 minus Theta
Let's derive the formula for sec 90 degrees minus theta.
Sec 90° – Theta = sec(90° – θ)
= sec(1 × 90° – θ)
Here, 90° is multiplied by 1, an odd number so sec will change to cosec. Also, 90° – θ comes in the first quadrant, which means all trigonometric ratios are positive.
So, sec(90° – θ) = cosec θ
Therefore, Sec 90 – Theta is equal to Cosec theta.
Similarly, we can derive the formula for sec 90 degrees plus theta.