# How to use the SEC function

**What is the SEC function?**

The SEC function calculates the secant of an angle.

## 1. Introduction

**What is a secant?**

The trigonometric secant is a function that defines an angle of a right-angled triangle to the ratio of the hypotenuse to the adjacent side. It is also the inverse of the cosine, sec(θ) = 1/cos(θ).

**What is the angle θ?**

The Greek letter theta (θ) is commonly used to represent an unknown angle in a right triangle.

**What is a right triangle?**

A right triangle is a type of triangle that contains one internal angle measuring 90 degrees or π/2 radians (a right angle).

**What are radians?**

Radians are a unit used to measure angles. An angle of 1 radian has an arc length equal to the circle's radius.

**What is the relationship between the number pi and radians?**

Radians measure angles by the length of the arc they make in a circle rather than degrees. The full circumference of any circle is 2π multiplied by the circle's radius (2πr).

Since the circumference goes all the way around a circle, that means the full circle measures 2π radians. Half a circle would be π radians (half of 2π). A quarter circle is 2π/4 = π/2 radians. An eighth of a circle is 2π/8 = π/4 radians.

Excel has a function that returns the number pi: PI function

**What is an arc?**

An arc is a curved segment of a circle's circumference, it is a portion of the circle's curve, defined by two endpoints.

In other words, an arc is formed by two radii intersecting the circumference and the enclosed edge between them.

**What is a circle's radius?**

The radius of a circle is the distance from the center point to any point on the circle's edge or circumference.

**What is radii?**

The plural form of the word "radius".

**What is the Pythagorean theorem?**

The Pythagorean theorem is a mathematical relationship between the sides of a right triangle. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

a^{2} + b^{2} = c^{2}

a and b are the lengths of the legs of the triangle

c is the length of the hypotenuse

For example, in a triangle with legs 5 and 2:

√(5^{2} + 2^{2)} = √(25 + 4) = √29

The hypotenuse is √29

**What are the main trigonometric functions?**

Function |
Domain (input) |
Range (output) |

sin(x) | All real numbers | (-1, 1) |

cos(x) | All real numbers | (-1, 1) |

tan(x) | All real numbers except multiples of π/2 | (-∞, ∞) |

sec(x) | All real numbers except multiples of π | (1, ∞) U (-∞, -1) |

csc(x) | All real numbers except integer multiples of π | (-∞, -1) U (1, ∞) |

cot(x) | All real numbers except integer multiples of π | (-∞, ∞) |

## 2. Syntax

SEC(*number*)

number |
Required. The number (radians) for which you want to calculate the secant. |

The secant is calculated like this:

## 3. Example 1

*Calculate the secant of **π/3 radians in a right triangle**?*

The argument is:

- number: C18 which contains π/3 radians or 1.0471975511966 radians.

Formula in cell C20:

The formula in cell C20 returns 2 which represents the secant in a right triangle based on an angle of π/3 or 60 degrees. The secant is the ratio between the hypotenuse and the adjacent side (b).

*Determine the hypotenuse (c) if the adjacent side (b) is equal to 1 unit and the angle (A) is equal to **π/4 **radians?*

Secant A = c/b

Secant π/3 = 2

2 = c/1

If the ratio 2 is equal to c/b then 2 = c/1

c = 2*1

c = 2

The hypotenuse is 2 units if angle A is π/3 (60 degrees) and the adjacent side (b) is 1 unit.

## 4. Example 2

*In a right triangle, if one acute angle measures 30 degrees and the adjacent is 10 units long, find the length of the hypotenuse?*

What we know:

- The hypotenuse is unknown.
- Angle A is 30 degrees. We need to convert degrees to radians.
- The adjacent side (b) is 10 units.
- SEC A = c/b (hypotenuse / adjacent side)

Hypotenuse = b * SEC A

Formula in cell C20:

The result in cell C20 is 11.5470053837925 Here is how it is calculated:

- Convert degrees to radians. RADIANS(30) equals 0.523598775598299
- Calculate the secant. SEC(RADIANS(30) equals 1.15470053837925
- Multiply the adjacent side to the ratio to get the length of the hypotenuse.

10*SEC(RADIANS(30)) equals 11.5470053837925

The RADIANS function converts degrees to radians.

## 5. Example 3

*Calculate the secant ratio if the adjacent side is 3 and the opposite side is 4 in a right triangle?*

What we know:

- Right triangle
- The adjacent side (b) is 3.
- The opposite side (c) is 4.
- The Pythagorean theorem allows us to calculate the hypotenuse.

a^{2}+ b^{2}= c^{2} - The secant ratio is hypotenuse / adjacent side. SEC A = c/b

The hypotenuse (c) is unknown. We can use the Pythagorean theorem to calculate the hypotenuse. Formula in cell C18:

The result is 5 which represents the length of the hypotenuse. Here is how it is calculated:

- Square the adjacent side. 3^2 = 9
- Square the opposite side . 4^2 = 16
- Add the squared values. 9 + 16 = 25
- Calculate the square root of the sum. 25^0.5 = 5

We now know that the hypotenuse is 5. All sides in the right triangle is now known. The secant ratio is hypotenuse / adjacent side which becomes 5 / 3.

Cell C20 displays 5/3. Cell C23 shows the angle A in radians. Formula in cell C23:

Cell C23 returns 0.927295218001612 radians which roughly corresponds to 53.13 degrees.

## 6. Example 4

**Find the length of the adjacent side, in a right triangle, if the hypotenuse is 7 units and angle A is π/4 radians?**

What we know:

- A right triangle meaning one of the angles (
**C**) is π/2 radians or 90 degrees. - Angle
**A**is π/4 radians or 45 degrees. This means that the adjacent and opposite sides are equal in length. - The hypotenuse is 7 units.

Formula in cell C21:

The formula in cell C21 returns 4.94974746830583 which represents the length of adjacent side (b). Here is how it is calculated:

- Calculate the secant using π/4 radians. SEC(PI()/4) equals 1.41421356237309 or √2
- Divide the length of the hypotenuse by the secant ratio. 7/√2 equals approx. 4.94

You can also calculate the adjacent side by using the Pythagorean theorem: a^{2} + b^{2} = c^{2}

The angle is π/4 radians or 45 degrees which makes the adjacent side and the opposite equal in length. x^{2} + x^{2} = c^{2}

2x^{2} = c^{2
}x^{2} = c^{2} / 2

x = √(c^{2} / 2)

x = √(7^{2} / 2)

x = √(49 / 2)

x = √24.5

x = 4.94974746830583

### Functions in 'Math and trigonometry' category

The SEC function function is one of 61 functions in the 'Math and trigonometry' category.

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