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Secant Square X Formula is a trigonometric function that returns the square of the secant function value for an angle x. It is denoted as sec2 x in Trigonometry.
- Secant Function is the ratio of the hypotenuse to the base in a right-angled triangle.
- It is abbreviated as sec x, where x is the acute angle.
- It is the reciprocal of cosine function and is thus written as sec x = 1/cos x.
- Sine, Cosine, Tangent, Secant, Cotangent, and Cosecant, are the six trigonometric functions.
Secant Square x Formula is given as:
| Sec 2 x = 1 + Tan2 x |
Read More: NCERT Solutions for Class 11 Mathematics Trigonometric Functions
Key Terms: Secant Square X Formula, Secant, Trigonometry, Right-Angled Triangle, Cosine, Tangent, Trigonometric Functions
What is Secant Function?
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Secant Function is a periodic function in trigonometry which is commonly abbreviated as ‘sec’.
- Secant Function is the ratio of the length of the hypotenuse to that of the length of the base in a right-angled triangle.
- It can also be defined as the reciprocal of the Cosine Function and is given as sec x = 1 / cos x.
Secant Function
If θ is the angle between the base and hypotenuse of a right-angled triangle then,
| Sec θ = Hypotenuse/Base = 1/Cos θ |
Where
- Hypotenuse is the longest side of the right-angled triangle.
- Base is the side adjacent to the angle.
- θ is the acute angle formed between the hypotenuse and the base.
Trigonometric Functions Detailed Video Explanation
Trigonometric Ratios
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Trigonometric Ratios are defined as the ratios of the length of sides of a right-angled triangle in Trigonometry. Trigonometric Ratios describe the relationship between the sides of a right triangle to the respective angle.
There are six Trigonometric Ratios in Trigonometry which are as follows:
- Sine (sin)
- Cosine (cos)
- Tangent (tan)
- Cotangent (cot)
- Secant (sec)
- Cosecant (cosec)
Sine (sin), Cosine (cos), and Tangent (tan) are the primary trigonometric ratios whereas Cotangent (cot), Secant (sec), and Cosecant (cosec) are derived from sin, cos, and tan respectively.
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Secant Square X Formula
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Secant Square x Formula is a mathematical formula in Trigonometry that returns the square of the secant function value for an angle x.
- Secant Square x Formula is expressed as sec2 x.
- The period of the function sec x is 2π, however, the period of sec2 x is π.
- It is equal to the sum of one and the tangent square function.
Secant Square x Formula is given as follows:
| Sec 2 x = 1 + Tan2 x |
Where
- x refers to one of the angles of the right-angled triangle.
- Tan x refers to the tangent function for angle x.
Read More: Trigonometric Functions Important Questions
Secant Square X Formula Derivation
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Secant and Tangent functions are represented in trigonometry as sec θ and tan θ, respectively. According to the Pythagorean identity of Secant and Tangent functions, they can be represented in the following manner in trigonometry:
| Sec2θ – Tan2θ = 1 |
Therefore, it can be rearranged and written as:
| Sec2θ = 1 + Tan2θ |
Thus, the secant square x formula is derived and proven in this manner. It can be stated that the secant square function is equal to the summation of one and the tangent square function.
Secant Square x Formula can also be derived using the identity of the sum of squares of sine and cosine ratios.
We know that,
| sin2 x + cos2 x = 1 |
Dividing both sides by cos2 x,
(sin2 x/cos2 x) + (cos2 x/cos2 x) = 1/cos2 x
tan2 x + 1 = sec2 x
sec2 x = 1 + tan2 x
Thus, the Secant Square x Formula is derived.
Secant Square X Formula Examples
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Here are some solved examples on Secant Square X Formula for a better understanding of the formula:
Example 1: Find the value of sec square x, if tan x is given as 2/5.
Solution: Tan x is given as 2/5.
Using the Secant Square X Formula,
Sec2x = 1 + tan2x
= 1 + (2/5)2
= 1 + 4/25
= (25 + 4) / 25
= 29/25
Thus, Sec2x is equal to 29/25 if tan x is given as 2/5.
Example 2: What will be the value of sec2x, if the value of cot x is 2/5?
Solution: It is given that cot x = 2/5.
tan x = 1/cot x = 5/2
Now, Using the Secant Square X Formula,
sec2x = 1 + tan2x
= 1 + (5/2)2
= 1 + 25/4
= 29/4
Thus, Sec2x is equal to 29/4 when cot x is 2/5.
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Secant Function Values
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The value of the Secant Function for standard angles such as 0°, 30°, 45°, 60°, and 90° along with other angles like 180°, 270°, and 360° are given below:
| Secant Function | Value |
|---|---|
| sec 0° | 0 |
| sec 30° | 2/√3 |
| sec 45° | √2 |
| sec 60° | 2 |
| sec 90° | Not Defined |
| sec 120° | -2 |
| sec 150° | -2/√3 |
| sec 180° | -1 |
| sec 270° | Not Defined |
| sec 360° | 1 |
These values can help in faster calculations involving trigonometric ratios.
Things to Remember
- Secant Function is an important trigonometric function abbreviated as "sec".
- Secant is the ratio of the hypotenuse to the base in a right-angled triangle.
- Secant Formula is given as sec θ = Hypotenuse/Base = 1/cos θ.
- Secant Square X Formula is used to find the square of the secant function value for an angle x.
- It is the summation of one and the tangent square function.
- Secant Square X Formula is given as Sec2x = 1 + tan2x.
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Sample Questions
Ques. In a right-angled triangle, find the value of sec θ if tan θ is given as 3. (3 Marks)
Ans. Given that, Tan θ = 3.
Using the Secant Square X Formula,
sec2x = 1 + tan2x
sec2x = 1 + 32
sec2x = 1 + 9 = 10
Sec x = √10
Thus, the value of sec θ, if tan θ is given as 3, will be √10.
Ques. Using a Secant Formula, find Sec x if Cos x equals 4/5. (2 Marks)
Ans. It is known that Secant Function is the reciprocal of the Cosine Function. Thus,
Sec x = 1 / Cos x
Sec x = 1 / (4/5)
Sec x = 5 / 4.
As a result, Sec x will have a value of 5/4 if Cos x equals 4/5.
Ques. Calculate the secant value of the angle in a right triangle with a hypotenuse of 13 units and an adjacent side of 5 units. (3 Marks)
Ans. Secant Function is defined as the ratio of the hypotenuse to the base in a right-angled triangle.
Thus,
Sec θ = Hypotenuse/Base
Substituting the values, we get
Sec θ = 13/5
As a result, the value of the secant function is 13/5.
Ques. What is the value of Sec Square x, if tan x is 1/3? (2 Marks)
Ans. Tan x is given as 1/3.
Using the Secant Square X Formula,
sec2x = 1 + tan2x
sec2x = 1 + (1/3)2
sec2x =1 + 1/9
= 10/9
Thus, sec2x = 10/9 if tan x is 1/3.
Ques. Simplify: (1 + tan2A) (1 – sin2A). (3 Marks)
Ans. The given expression is (1 + tan2A) (1 – sin2A).
Now, according to the Secant Square X Formula,
sec2x = 1 + tan2x
And, cos2A = 1 – sin2A
Substituting the values, we get
= (1 + tan2A)(1 – sin2A)
= sec2A . cos2A
= (1/cos2A) x cos2A
= 1
Thus, (1 + tan2A) (1 – sin2A) is equal to 1.
Ques. What is the value of Sec Square x, if tan x is 4/5? (2 Marks)
Ans. Tan x is given as 4/5.
Using the Secant Square X Formula,
sec2x = 1 + tan2x
= 1 + (4/5)2
= 1 + 16/25
= 41/25
Thus, sec2x = 41/25 when tan x is 4/5.
Ques. Simplify (1 + tan2A) (1 – sin A) (1 + sin A). (3 Marks)
Ans. The given expression is (1 + tan2A) (1 – sin A) (1 + sin A).
Using the (a+b) (a-b) = a2 – b2 identity,
(1 – sin A) (1 + sin A) = 1 – sin2A
Now, substitute this value in the given expression,
= (1 + tan2A)(1 – sin2A)
= sec2A x cos2A (As sec2x = 1 + tan2x, and cos2A = 1 – sin2A)
= (1/cos2A) x cos2A
= 1
Thus, (1 + tan2A) (1 – sin A) (1 + sin A) = 1.
Ques. Calculate the base of a right-angled triangle with a 20-unit hypotenuse and a 30-degree base angle. (3 Marks)
Ans. Given that,
- θ = 30 Degrees
- Hypotenuse = 20 units
Using Secant Formula,
Sec θ = Hypotenuse/Base
Sec 30 =20/B
2/√3 = 20/B (As Sec 30 = 1/√3)
B = (20 x √3) / 2
B = 10√3
As a result, a right-angle triangle's base side is 10√3 Units.
Ques. If the hypotenuse is 13 units, the base is 5 units, and the perpendicular is 12 units, find sec θ using the secant formula. (3 Marks)
Ans. Given that,
- Perpendicular P = 12
- Base B = 5
- Hypotenuse H = 13
Using Secant Formula,
Sec θ = Hypotenuse/Base
sec θ = 13/5
sec θ = 2.6
Therefore, sec θ is 2.6
Ques. Using Secant Formula, find Sec θ if Cos θ is given as 3/5. (2 Marks)
Ans. It is given that Cos θ = 3/5
Now, it is known that Secant Function is the reciprocal of the Cosine Function. Thus,
sec θ = (1/cos θ)
sec θ = 1/(3/5)
sec θ = 5/3
Therefore, sec θ is 5/3 if Cos θ = 3/5.
Ques. Prove that: (Sec X Sec Y + Tan X Tan Y )2 − ( Sec X Tan Y + Tan X Sec Y )2 = 1. (5 Marks)
Ans. LHS = (Sec X Sec Y + Tan X Tan Y )2 − ( Sec X Tan Y + Tan X Sec Y )2
= [(sec x sec y)2 + (tan x tan y)2 – 2(sec x sec y)(tan y x tan y)] - [(sec x tan y)2 + (tan x sec y)2 – 2(sec x tan y)(tan y x tan y)]
= [sec2 x sec2 y + tan2 x tan2 y – 2sec x sec y tan y x tan y] - [sec2 x tan2 y + tan2 x sec2 y – 2sec x tan y tan y x tan y]
= sec2x sec2 y + tan2x tan2 y – 2sec x sec y tan y x tan y - sec2 x tan2y + tan2 x sec2 y + 2sec x tan y tan y x tan y
= sec2x sec2 y + tan2x tan2y - sec2x tan2y + tan2 x sec2 y
= sec2x(sec2 y - tan2y) + tan2x(tan2 y - sec2 y)
= sec2x(sec2 y - tan2y) - tan2 x(sec2 y - tan2 y)
= sec2x(1) - tan2 x(1)
= sec2 x - tan2 x
= 1
RHS = 1
LHS = RHS
Hence Proved.
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