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.
a2 + b2 = c2
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:
√(52 + 22) = √(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.
a2 + b2 = c2 - 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: a2 + b2 = c2
The angle is π/4 radians or 45 degrees which makes the adjacent side and the opposite equal in length. x2 + x2 = c2
2x2 = c2
x2 = c2 / 2
x = √(c2 / 2)
x = √(72 / 2)
x = √(49 / 2)
x = √24.5
x = 4.94974746830583
Functions in 'Math and trigonometry' category
The SEC function function is one of 62 functions in the 'Math and trigonometry' category.
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