Secant Formula – Concept, Formulae, Solved Examples

A vital branch of mathematics that is concerned with the relationships between the angles and lengths of a right triangle is referred to as trigonometry. Sine, cosine, tangent, cosecant, secant and cotangent are the six trigonometric ratios, and a trigonometric ratio is the ratio of the sides of a right-angled triangle. Since cosecant, secant, and cotangent functions are the reciprocal functions of sine, cosine, and tangent functions, respectively, sine, cosine, and tangent functions are the three significant trigonometric functions.

The six trigonometric ratios or functions are,

1. sin θ = Opposite side/Hypotenuse
2. cos θ = Adjacent side/Hypotenuse
3. tan θ = Opposite side/Adjacent side
4. cosec θ = 1/sin θ = Hypotenuse/Opposite side
5. sec θ = 1/cos θ = Hypotenuse/Adjacent side 
6. cot θ = 1/tan θ = Adjacent side/Opposite side

Secant Formula

The 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 

secθ = Hypotenuse/Base

Some Basic Secant Formulae

Some basic trigonometric formulae in terms of other trigonometric formulae are discussed below

Secant Function in Quadrants

• The 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)

The negative angle identity of a secant function

• The secant of a negative angle is always equal to the secant of the angle.

sec (-θ) = sec θ

Secant function in terms of the Cosine function

• A secant function is a reciprocal function of the cosine function.

sec θ = 1/cos θ

Secant function in terms of the Sine function

The 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 the 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 the Tangent function

The secant function in terms of the tangent function can be written as,

sec θ = ±√(1 + tan²θ)

From the Pythagorean identities, we have,

sec² θ – tan² θ = 1

⇒ sec²θ = 1 + tan²θ

Hence, sec θ = ±√(1 + tan²θ)

Secant function in terms of the 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 the Cotangent function

The secant function in terms of the cotangent function can be written as,

sec θ = ±√(cot²θ + 1)/cotθ

From the 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

Angle (In degrees)	Angle (In Radians)	cos θ	sec θ = 1/cosθ
0°	0	1	1/1 = 1
30°	π/6	√3/2	1(√3/2) = 2/√3
45°	π/4	1/√2	1/(1/√2) = √2
60°	π/3	1/2	1/(1/2) = 2
90°	π/2	0	1/0 = undefined
180°	π	-1	1/-1 = -1

Sample Problems

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.