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Sec 0

 

The word secant comes from the Latin word secare, meaning to cut. So secant is also referred to as the intersecting lines inside a circle. But that is another case. Here, we are referring to the secant angle in trigonometry.Secant of any angle or sec x is defined as the ratio of the length of hypotenuse to the adjacent side of the angle for which secant angle is to be found out. Secant is the inverse of cosine (generally known as cos). Secant angle is positive in the 4th quadrant. Sec 0 = 1/cos 0

Graph of a secant function

Since secant is the inverse of the cosine function, therefore, it is infinity wherever the cosine function is zero.

Derivative of secant function

The derivative of a secant function is sec(x) tan (x).

According to trigonometry, the value of sec 0 is 1. 

Proof – 

Consider a right-angled triangle PQR with angle P as the secant angle for which we need to find sec 0. 

According to the secant definition, sec of angle P = length of the hypotenuse/length of the adjacent side 

= PQ/PR 

Since angle P will be zero, this implies that the value of the length PQ will be equal to the length of PR. 

Therefore, sec 0 = PR/PR, which gives the value of sec 0 = 1. 

Hence, proved. 

Also, cos 0 is 1. The inverse of cos 0 will be 1. 

Sec 0 = 1/cos 0 = 1/1 = 1 

Inverse of a secant function

The inverse of a secant function is known as the arcsec function. Whenever we see arcsec A written, we can infer the angle whose secant is A. 

arcsec x = sec-1

sec 60 = 2.000 Means: The secant of 60 degrees is 2.000

arcsec 2.0 = 60 Means: The angle whose secant is 2.0 is 60 degrees.
 

Important formulas of secant angles-

sec 2 y = 1 + tan 2

sec(90°–x) = x

sec(2x) = x/(2–x) 
 

Fun fact

  • Trigonometry is the most used concept of mathematics in astronomy and surveying.
  • The secant method is trendy in making large complex buildings. This method uses the roots of secant lines to find an approximate root of a particular function.
  • Trigonometry also finds its usage in navigation, aviation, criminology, marine biology, and many more branches of science.

Example

Find the secant angle of W. 

Solution 

According to the definition, sec W = hypotenuse / base of angle W

Here, hypotenuse is WV = 37 (hypotenuse is always the opposite line of the right angle) 

Base of angle W is WU = 12 

Therefore, sec W = 37/12 

Special case

If the value of angle W is 0, then there will be no triangle. Only the straight line WU will be present. This implies that the value of sec 0 will be 1.

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