Secant Formula - Concept, Formulae, Solved Examples

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,

1. sin θ = (Opposite Side/Hypotenuse = AB/AC
2. cos θ = Adjacent Side/Hypotenuse = BC/AC
3. tan θ = Opposite side/adjacent side = AB/BC
4. cosec θ = 1/sin θ = Hypotenuse/Opposite Side = AC/AB
5. sec θ = 1/cos θ = Hypotenuse/Adjacent Side = AC/BC
6. cot θ = 1/tan θ = Adjacent Side/Opposite Side = BC/AB

Secant Formula

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.

An 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

Basic Secant Formulae

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.

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:

Sample Problems on Secant Formula

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.