Secant Formula - Concept, Formulae, Solved Examples
Last Updated : 22 Jul, 2024
Secant is one of the six basic trigonometric ratios and its formula is secant(θ) = hypotenuse/base, it is also represented as, sec(θ). It is the inverse(reciprocal) ratio of the cosine function and is the ratio of the Hypotenus and Base sides in a right-angle triangle.
In this article, we have covered, about Scant Formula, related examples and others in detail.
What are Trigonometric Ratios?
Trigonometric ratios are ratios of sides in a triangle and there are six trigonometric ratios. In a right-angle triangle, the six trigonometric ratios are defined as:
The six trigonometric ratios or functions are,
- sin θ = (Opposite Side/Hypotenuse = AB/AC
- cos θ = Adjacent Side/Hypotenuse = BC/AC
- tan θ = Opposite side/adjacent side = AB/BC
- cosec θ = 1/sin θ = Hypotenuse/Opposite Side = AC/AB
- sec θ = 1/cos θ = Hypotenuse/Adjacent Side = AC/BC
- cot θ = 1/tan θ = Adjacent Side/Opposite Side = BC/AB
Secant of an angle in a right-angled triangle is the ratio of the length of the hypotenuse to the length of the adjacent side to the given angle. We write a secant function as "sec". Let PQR be a right-angled triangle, and "θ" be one of its acute angles.
Secant FormulaAn adjacent side is a side that is adjacent to the angle "θ", and a hypotenuse is a side opposite to the right angle and also the longest side of a right-angled triangle. A secant function is a reciprocal function of the cosine function.
Now, the secant formula for the given angle "θ" is,
sec θ = Hypotenuse/Adjacent side
or
sec θ = Hypotenuse/Base
Some basic trigonometric formulae in terms of other trigonometric formulae are discussed below
Secant Function in Quadrants
- Secant function is positive in the first and fourth quadrants and negative in the second and third quadrants.
Degrees | Quadrant | Sign of Secant function |
|---|
0° to 90° | 1st quadrant | + (positive) |
90° to 180° | 2nd quadrant | – (negative) |
180° to 270° | 3rd quadrant | – (negative) |
270° to 360° | 4th quadrant | + (positive) |
Negative Angle Identity of a Secant Function
- Secant of a negative angle is always equal to the secant of the angle.
sec (-θ) = sec θ
Secant Function in terms of Cosine Function
- A secant function is a reciprocal function of the cosine function.
sec θ = 1/cos θ
Secant Function in terms of Sine Function
Secant function in terms of the sine function can be written as,
sec θ = ±1/√(1-sin2θ)
We know that
sec θ = 1/cos θ
From Pythagorean identities we have;
cos2 θ + sin2 θ = 1
⇒ cos θ = √1 - sin2 θ
Hence, sec θ = ± 1/√(sin2 θ - 1)
Secant Function in terms of Tangent function
The secant function in terms of the tangent function can be written as,
sec θ = ±√(1 + tan2θ)
From Pythagorean identities, we have,
sec2 θ – tan2 θ = 1
⇒ sec2θ = 1 + tan2θ
Hence, sec θ = ±√(1 + tan2θ)
Secant Function in terms of Cosecant Function
The secant function in terms of the cosecant function can be written as,
If θ is positive in the first quadrant, then
sec θ = cosec (90 - θ) or cosec (π/2 - θ)
(or)
sec θ = cosec θ/√(cosec2 θ - 1)
We have,
sec θ = 1/√(1-sin2θ)
We know that sin θ = 1/cosec θ
By substituting sin θ = 1/cosec θ in the above equation, we get
sec θ = 1/√(1 - (1/cosec2θ)
Hence, sec θ = (cosec θ)/√(cosec2 θ - 1)
Secant Function in terms of Cotangent Function
The secant function in terms of the cotangent function can be written as,
sec θ = ±√(cot2θ + 1)/cotθ
From Pythagorean identities, we have,
sec2 θ – tan2 θ = 1
⇒ sec2θ = 1 + tan2θ
We know that tan θ = 1/cot θ
By substituting tan θ = 1/cot θ in the above equation, we get
⇒ sec2 θ = 1 + (1/cot2θ)
⇒ sec2 θ = (cot2 θ + 1)/cot2θ
Hence, sec θ = ±√(cot2θ + 1)/cotθ
Trigonometric Ratio Table
The trigonometric table is added below:
Problem 1: Find the value of sec θ, if sin θ = 1/3.
Solution:
Given,
sin θ = 1/3
We know that,
sec θ = 1/√(1-sin2θ)
⇒ sec θ = 1/(1 - (1/3)2)
= 1/√(1 - (1/9))
= 1/√(8/9) = 3/2√2
Hence, sec θ = 3/2√2
Problem 2: Find the value of sec x if tan x = 5/12 and x is the first quadrant angle.
Solution:
Given,
tan x = 5/12
From the Pythagorean identities, we have,
sec2 x – tan2 x = 1
⇒ sec2x = 1 + tan2x
⇒ sec2x = 1 + (5/12)2
⇒ sec2x = 1 +(25/144) =169/144
⇒ sec x = √(169/144) = ±13/12
Since x is the first quadrant angle, sec x is positive.
Hence, sec x = 13/12
Problem 3: If cosec α = 25/24, then find the value of sec α.
Solution:
Given,
cosec α = 25/24
We know that,
cosec α = 25/24 = hypotenuse/opposite side
adjacent side = √[(hypotenuse)2 - (opposite side)2]
= √[(25)2 - (24)2] = √(625 - 576)
= √49 = 7
Now, sec α = hypotenuse/adjacent side = 25/7
Hence, sec α = 25/7
Problem 4: Find the value of sec θ, if cos θ = 2/3.
Solution:
Given,
cos θ = 2/3
We know that,
A secant function is the reciprocal function of a cosine function.
So, sec θ = 1/cos θ
= 1/(2/3) = 3/2
Hence, sec θ = 3/2
Problem 5: A right triangle has the following measurements: hypotenuse = 10 units, base = 8 units, and perpendicular = 6 units. Now, find sec θ using the secant formula.
Solution:
Given,
Hypotenuse = 10 units
Base = 8 units
Perpendicular = 6 units
We know that,
sec θ = hypotenuse/base
= 10/8 = 5/4
Hence, sec θ = 5/4.
Problem 6: Determine the side of a right-angled triangle whose hypotenuse is 15 units and whose base angle with the side is 45 degrees.
Solution:
Given,
θ = 45 degree
Hypotenuse = 15 units
Using the secant formula,
sec θ = hypotenuse/base
sec 45 =15/B
√2 = 15/B
B = 15/√2 = 15√2/2
B = 7.5√2
Hence, the base of the triangle is 7.5√2 units.
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