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.

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 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

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

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.

What is the rule for secant?

Rule for secant states that secant is equal to the ratio of the hypotenuse and base and is resiprocal of cosine function and is represented as, sec x = 1/cos x.

How to calculate the sec?

Sec is calculated using the formula, sec A = Hypotenuse / Base

Is secant equal to sin?

No, secant is not equal to sin.