# 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,

- sin θ = Opposite side/Hypotenuse
- cos θ = Adjacent side/Hypotenuse
- tan θ = Opposite side/Adjacent side
- cosec θ = 1/sin θ = Hypotenuse/Opposite side
- sec θ = 1/cos θ = Hypotenuse/Adjacent side
- 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° | 1 | + (positive) |

90° to 180° | 2 | – (negative) |

180° to 270° | 3 | – (negative) |

270° to 360° | 4 | + (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-sin^{2}θ)We know that

sec θ = 1/cos θ

From the Pythagorean identities we have;

cos

^{2}θ + sin^{2}θ = 1⇒ cos θ = √1 – sin

^{2}θHence, sec θ = ± 1/√(sin

^{2}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 + tan^{2}θ)From the Pythagorean identities, we have,

sec

^{2}θ – tan^{2}θ = 1⇒ sec

^{2}θ = 1 + tan^{2}θHence, sec θ = ±√(1 + tan

^{2}θ)

**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/√(cosec^{2}θ – 1)We have,

sec θ = 1/√(1-sin

^{2}θ)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/cosec

^{2}θ)Hence, sec θ = (cosec θ)/√(cosec

^{2}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 = ±√(cot^{2}θ + 1)/cotθFrom the Pythagorean identities, we have,

sec

^{2}θ – tan^{2}θ = 1⇒ sec

^{2}θ = 1 + tan^{2}θWe know that tan θ = 1/cot θ

By substituting tan θ = 1/cot θ in the above equation, we get

⇒ sec

^{2}θ = 1 + (1/cot^{2}θ)⇒ sec

^{2}θ = (cot^{2}θ + 1)/cot^{2}θHence, sec θ = ±√(cot

^{2}θ + 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-sin^{2}θ)⇒ 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,

sec^{2}x – tan^{2}x = 1⇒ sec

^{2}x = 1 + tan^{2}x⇒ sec

^{2}x = 1 + (5/12)^{2}⇒ sec

^{2}x = 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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