Content Curator
The ratio of the length of the hypotenuse to the length of the neighboring side is the sec of an angle in a right triangle. In other terms, secant is the cosine function's inverse. Trigonometry establishes the relation between a right triangle's sides and angles. The sine and cosine of an angle are the two basic trigonometric ratios. These two basic trigonometric functions can be used to define the other four trigonometric ratios. The ratio of the hypotenuse to the adjacent side of the reference angle is known as the secant of an angle.
- Sec θ = Hypotenuse / Adjacent side
- Value of Sec 90° = ∞
- The value of Sec 90 (in radians) = -2.2317761286…
Check Important Notes for Class 10 Triangles
| Table of Content |
Value of Sec 90º
A quotient obtained by dividing the hypotenuse of a right triangle by the side nearest to the angle whose secant is to be calculated is the secant of an angle. The transposition or reciprocal of any angle's cosine is the secant. As a result, Sec = 1/ Cos can be used to calculate the secant and cosine of an angle. This is the reference angle for which the trigonometric ratio must be calculated. The cosine of an angle of 90º degrees is zero. As a result, the secant of a 90º angle can be expressed as 1/0. When the divisor is zero, the quotient is undefined, hence it is referred to as.
Value of Sec 90º
Read Also: Trigonometric Identities
Calculation of Sec 90º
The concept of trigonometric ratios of standard angles assists in calculating the value of sec 90 degrees. A right triangle is used to obtain the trigonometric ratios of the standard angle 90°.
Consider the right triangle in the illustration above. Assume that A is the reference angle. If angle A is to be increased to 90º, the hypotenuse must be transferred to the perpendicular or opposite side, as seen in the following photos of the previous figure. We can see that when the reference angle is equal to 90º, the base or neighboring side is equal to zero. In relation to the reference angle, we also notice that the hypotenuse overlaps with the perpendicular or opposite side. As a result, the length of the opposing side, or perpendicular, is equal to the triangle's hypotenuse. Mathematically,
Base / Adjacent Side = 0 and Perpendicular / Opposite side = Hypotenuse
By definition of the cosine of an angle θ is given as:
Cos θ = Adjacent side/Hypotenuse
The secant of an angle θ is the reciprocal of its cosine.
Sec θ = 1/Cosθ
Sec θ =1/Adjacent side/Hypotenuse
Sec θ = Hypotenuse/Adjacent side
Sec 90° = Hypotenuse0
Regardless of the length of the triangle's hypotenuse, the secant of the angle 90º is not determined since the quotient is equal to infinity as long as the base or neighboring side is equal to 0.
Also Read:
Sec Inverse x Formulas
Now that we've looked at the concept of sec inverse x, let's look at some sec inverse x formulas that can be utilised to solve a variety of mathematics trigonometric problems:
- sec-1(-x) = π - sec-1x
- sec-1x + cosec-1x = π/2
- cos-1x = sec-1(1/x)
- cos-1(1/x) = sec-1x
Points To Remember
- To memorize the definitions of the three basic trigonometric ratios, utilize the English phrase "Some People Have Curly Brown Hairs Turned Permanently Black."
- The reciprocal of a reference angle's cosine is its secant. It should never be mistaken with its cosine's inverse. Inverse trigonometric functions are not the same as regular trigonometric functions.
- Secant is derived from the Latin word "secare", which means "to cut." A secant intersects a circle at exactly two locations in the case of a circle.
- The length of the hypotenuse divided by the length of the neighboring side is the secant of an angle.
- The reciprocal of the cosine is the secant. In a right triangle, it is the ratio of the hypotenuse to the side adjacent to a certain angle.
Also Check:
Sample Questions
Ques 1. What is the value of Sec(270 – x) Sec(90 – x) – tan (270 – x) tan (90 – x)? (2 marks)
Ans. Sec(270 – x) Sec(90 – x) – tan (270 – x) tan (90 – x)=?
=( – cosec x cosec x) – ( – cot x cot x)
= -cosec 2 x + cot2 x
= – 1
Ques 2. Find the value of sin 30° + 1/sec 90°. (2 marks)
Ans. We know that, sec x = 1/cos x or cos x = 1/sec x
Thus, sin 30° + 1/sec 90° = sin 30° + cos 90°
= 1/2 + 0
= ½
Also Read:
Ques 3. How to Calculate Sec 90º and the Values of Other Trigonometric Ratios of 90° and 0°? (5 marks)
Ans. A right triangle can be thoroughly examined to find the trigonometric ratios of 0° and 90°.
Let angle A be the reference angle in the right triangle ABC illustrated above. The position of C should be changed to B in order for A to equal 0. As a result, when the reference angle is 0°, the hypotenuse length is zero and the base equals the hypotenuse. Similarly, the position of A should be shifted to B in order for the reference angle to be equal to 90°. The base will be 0 in this situation, and the perpendicular will be equal to the hypotenuse.
As a result, the trigonometric ratios of 0° and 90° are as follows:
| Trigonometric ratios | θ = 0º P = 0, H = B | θ = 90º 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 |
Read Also: Analytical Geometry
Ques 4. Sec(90−θ)= (3 marks)
Ans. In the figure,
sec(90−θ)= r/y =cscθ
So, sec(90−θ)=cscθ
So, cscθ is correct.
Ques 5. Is sec 90 degrees undefined? (1 mark)
Ans. The value of Secant 90 degrees is unknown in the trigonometric table and cannot be determined.
Ques 6. How do you solve 90º SEC? (1 mark)
Ans. As a result, you get sec(90°)=1cos(90°)=10, which is impossible to calculate. Using, say, the limit, you can get "near" 90 degrees from the left or right. Near 90°, you'll see that the sec tends to.
Ques 7. What is the value of Sec 90 in a fraction? (1 mark)
Ans. The cosine of an angle of 90° degrees is zero. As a result, the secant of a 90° angle can be expressed as 1/0.
Also Check:
Ques 8. What does Sec 90 theta equal? (1 mark)
Ans. sec (90° + θ) = - csc θ.
Ques 9. What is the value of Sec(270 – x) Sec(90 – x) – tan (270 – x) tan (90 – x)? (2 marks)
Ans. Sec(270 – x) Sec(90 – x) – tan (270 – x) tan (90 – x)=?
=( – cosec x cosec x) – ( – cot x cot x)
= -cosec 2 x + cot2 x
= – 1
Ques 10. Find the value of sin 30° + 1/sec 90°. (2 marks)
Ans. We know that, sec x = 1/cos x or cos x = 1/sec x
Thus, sin 30° + 1/sec 90° = sin 30° + cos 90°
= 1/2 + 0
= ½
Mathematics Related Links:
Comments