What Is the Value of Sec 90 Degrees and Why Is It Undefined
Value of Sec 900:
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.
What is Sec 90 value:
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.
How to Calculate Sec 90 Degrees?
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 900 = ∞
Examples on Sec 90 value:
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 ∞) = ∞
Fun Facts:
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.
FAQs on Sec 90 Degrees Explained with Concept and Value
1. What is the value of sec 90°?
The value of sec 90° is undefined because cos 90° equals 0. Since secant is the reciprocal of cosine, sec θ = 1 / cos θ. At θ = 90°:
- cos 90° = 0
- sec 90° = 1 / 0
- Division by zero is undefined
2. Why is sec 90° undefined in trigonometry?
Sec 90° is undefined because it involves division by zero. Using the identity sec θ = 1 / cos θ, and since cos 90° = 0, we get 1/0, which is mathematically undefined. Division by zero has no real value, so sec 90° does not exist in the real number system.
3. What is the formula for sec θ?
The formula for secant is sec θ = 1 / cos θ. Secant is the reciprocal trigonometric ratio of cosine. In a right triangle:
- cos θ = adjacent / hypotenuse
- sec θ = hypotenuse / adjacent
4. What happens to sec θ when cos θ equals zero?
When cos θ equals zero, sec θ becomes undefined. Since secant is the reciprocal of cosine:
- If cos θ = 0
- Then sec θ = 1 / 0
5. Is sec 90° infinity?
Sec 90° is not infinity; it is undefined. Although 1 divided by a very small number becomes very large, exactly at 90° we have cos 90° = 0, and 1/0 is undefined, not infinity. In limits, sec θ approaches infinity as θ approaches 90°, but at exactly 90°, it has no value.
6. How do you evaluate sec 90° using the unit circle?
Using the unit circle, sec 90° is undefined because the cosine value at 90° is zero. On the unit circle:
- The coordinate at 90° is (0, 1)
- Cosine equals the x-coordinate
- cos 90° = 0
7. What are the angles where sec θ is undefined?
Sec θ is undefined at angles where cos θ = 0. Cosine equals zero at:
- 90°
- 270°
8. What is the difference between sec 90° and cos 90°?
The difference is that cos 90° = 0, while sec 90° is undefined. Cosine directly gives the x-coordinate on the unit circle, which is zero at 90°. Secant is the reciprocal of cosine, so it becomes 1/0, which has no defined value.
9. Can you give an example showing why sec 90° is undefined?
Sec 90° is undefined because it results in division by zero. Example:
- Formula: sec θ = 1 / cos θ
- Substitute θ = 90°
- cos 90° = 0
- sec 90° = 1 / 0
10. Does sec 90° exist in real numbers?
Sec 90° does not exist in the real number system because it is undefined. Since sec θ = 1 / cos θ and cos 90° = 0, the expression becomes 1/0. Division by zero has no real value, so sec 90° is not a real number.
