Trigonometry is a branch of mathematics that deals with the study of measurements of sides and angles of a three-sided geometric figure (triangle). It establishes the relationship between the sides and angles of a right triangle. The two basic trigonometric ratios are sine and cosine of an angle. All the other four trigonometric ratios can be defined with these two basic trigonometric functions. In a right triangle, the side opposite to the right angle is the longest side and is called the hypotenuse. If any of the other two angles except the right angle is considered to be the reference angle, then the side that is adjacent to the reference angle is called the adjacent side or the base and the side opposite to the reference angle is called the opposite side of the perpendicular. The secant of an angle is defined as the ratio of the hypotenuse and the adjacent side of the reference angle.
The secant of any angle is described as a quotient obtained by dividing the hypotenuse of the right triangle by the side adjacent to the angle whose secant is to be determined. Secant of any angle is the transpose or the reciprocal of its cosine. So, secant and cosine of an angle can be related as Sec = 1/ Cos . Here θ is the reference angle whose trigonometric ratio is to be determined. The cosine of the angle equal to 900 is equal to 0. So, secant of the angle equal to 90o can be written as 1/0. The quotient of when the divisor is zero is undefined and hence it is considered as ∞. There is a detailed description of how to calculate Sec 90 value in the subsequent sections.
The concept of trigonometric ratios of standard angles helps in finding the value of sec 90 degrees. To find the trigonometric ratios of the standard angle 900, a right triangle is constructed.
Let us consider the right triangle shown in the above figure. Let ∠A be the reference angle. If angle A should become 900, then the hypotenuse should be shifted towards the perpendicular or the opposite side as shown in the subsequent images of the above figure. When the reference angle is equal to 900, we observe that the base or the adjacent side is equal to zero. We also observe that the hypotenuse overlaps with the perpendicular or the opposite side with respect to the reference angle. So, the length of the opposite side or the perpendicular is equal to the hypotenuse of the triangle. Mathematically,
Base / Adjacent Side = 0 and Perpendicular / Opposite side = Hypotenuse
By definition of the cosine of an angle θ is given as:
Cos θ = \[\frac{Adjacent side}{Hypotenuse}\]
The secant of an angle θ is the reciprocal of its cosine.
Sec θ = \[\frac{1}{Cos \theta}\]
Sec θ =\[\frac{1}{\frac{Adjacent side}{Hypotenuse}}\]
Sec θ = \[\frac{Hypotenuse}{Adjacent side}\]
Sec 900 = \[\frac{Hypotenuse}{0}\]
In Spite of the length of the hypotenuse of the triangle, the secant of the angle 900 is not defined because whatever may the length of the hypotenuse, the quotient is equal to infinity as long as the base or adjacent side is equal to 0.
Sec 90^{0} = ∞
Find the value of Cos 900 . Sec 900 - 1 / Sec 900. (Use Sec 90 value)
Solution:
Cos 900 . Sec 900 - 1 / Sec 900 = 1 x ∞ - 1/∞
= ∞ - 0
= ∞
Find the value of Sec-1 (Sec 900) and Sec (Sec-1 ∞)
Solution:
Sec 90 value = ∞
Secant and inverse secant are inverse operations with respect to each other.
So, Sec-1 (Sec 900) = 900
Sec (Sec-1 ∞) = ∞
The English statement “Some People Have Curly Brown Hairs Turned Permanently Black” can be used to remember the definitions of the three basic trigonometric ratios.
Secant of a reference angle is the reciprocal of its cosine. It should never be confused with the inverse of its cosine. Inverse trigonometric functions are entirely different.
1. How to Calculate Sec 90^{0} and the Values of Other Trigonometric Ratios of 90^{0} and 0^{0}?
The trigonometric ratios of 0^{0} and 90^{0} can be found by thoroughly analysing a right triangle.
In the right triangle, ABC shown above let the angle A be the reference angle. For A to be equal to 0, the position of C should be shifted to B. So, when the reference angle is 0^{0}, the length of the hypotenuse is zero and the base is equal to the hypotenuse. Similarly, for the reference angle to be equal to 90^{0}, the position of A should be shifted to B. In this case, the base will be equal to zero and the perpendicular is equal to the hypotenuse.
So, the trigonometric ratios of 0^{0} and 90^{0} are determined as follows:
Trigonometric ratios | θ = 0^{0} P = 0, H = B | θ = 90^{0} B = 0, H = P |
Sin θ = O / H | 0 | 1 |
Cos θ = A / H | 1 | 0 |
Tan θ = O / A | 0 | ∞ |
Cot θ = A / O | ∞ | 0 |
Sec θ = H / A | 1 | ∞ |
Cosec θ = H / O | ∞ | 1 |
2. How Can the Trigonometric Table for Standard Angle Be Remembered Easily?
There are several feasible tricks to remember and reciprocate the trigonometric table for standard angles easily. One such method is to write all the trigonometric values of standard angles just by remembering the first row i.e. the sine of standard angles.
The first row in the table is to be remembered which gives the values of the sine of the standard angles.
The second row which gives the cosine of the standard angles is written as elements in the first row written in reverse order.
The third row which gives the tangent of the angles is the ratio of the first two rows.
The fourth row is the cotangent which is the reciprocal of the elements of 3rd row.
The fifth row is the secant of the angles and is the reciprocal of the 2nd row,
The last row is the cosecant of the standard angles which is either the elements of the fifth row written in reverse order or the reciprocals of the elements in 1st row,
Trigonometric Ratios | 0^{0} | 30^{0} | 45^{0} | 60^{0} | 90^{0} |
Sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
Cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
Tan | 0 | 1/√3 | 1 | √3 | ∞ |
Cot | ∞ | √3 | 1 | 1/√3 | 0 |
Sec | 1 | 2/√3 | √2 | 2 | ∞ |
Cosec | ∞ | 2 | √2 | 2/√3 | 1 |