# How to use the ASIN function

**What is the ASIN function?**

The ASIN function calculates the arc-sine of a number. The angle is in radians and is between -pi/2 to pi/2.

#### Table of Contents

## 1. Introduction

**What is the SINE function?**

The sine is a trigonometric function that relates an angle θ in a right triangle to the ratio of the length of the side opposite the angle and the length of the longest side (hypotenuse) of the triangle. A right triangle has one angle that measures 90° or π/2 radians which is approximately 1.5707963267949 radians.

The SIN function calculates the ratio between the opposite side and the hypotenuse.

**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 the arc-sine of a number?**

The arc-sine is the inverse sine also written sin^{-1}. The inverse cotangent is used to find the angle θ when given the sine ratio.

The relationship between the sin function and the arc-sin function is as follows:

In a right-angled triangle, where:

A is the angle (in radians)

c is the length of the hypotenuse

a is the length of the opposite side

The sine of the angle A can be expressed as:

sine(A) = a/c

By taking the arc-sin (ASIN) of the ratio of a (opposite) and c (hypotenuse), we can find the angle A:

A = arc-sin (a/c)

This means that the ASIN function calculates the angle (in radians) when given the ratio of the opposite side to the hypotenuse. It is often displayed as SIN^{-1} in scientific calculators.

**What is the opposite side?**

The opposite side is the side opposite to the angle being considered. The image above shows a right-angled triangle, it has three internal angles represented by **A**, **B**, and **C**. The opposite side is determined by the chosen angle **A**, **B** or **C**. **A** has the opposite side a, **B** - b, and **C** - c

**What is the adjacent side?**

The adjacent side is the side that is in contact with the angle being considered and the right angle.

**What is the hypotenuse?**

The hypotenuse is the longest side of the right-angled triangle. It is the side opposite to the right angle (90 degrees).

**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 radius of a circle?**

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

**What are degrees?**

Degrees are a unit used to measure angles. It is based on dividing a full circle into 360 equal parts. Degrees are divided into fractional parts like minutes and seconds for more precision.

**What is the relationship between radians and degrees?**

The circumference of a circle is 360 degrees or 2π radians.

360 degrees = 2π radians

which is

degrees = radians x (180 / π)

Excel has two functions for converting between radians and degrees: RADIANS | DEGREES

## 2. Syntax

ASIN(*number*)

number |
Required. The number must be from -1 to 1. |

## 3. Example 1

**Find the angle A (in radians) which is located between the hypotenuse (c) and the adjacent side (b) of a right-angled triangle, where the opposite side is 4 units, and the hypotenuse is 5 units?**

**C** = π/2 radians (90°)

The argument is:

- number = adjacent / hypotenuse = a/c = 4/5 = 0.8

Formula in cell C20:

The formula in cell C20 returns 0.927295218001612 radians which represents the angle for A in the image above. To get the result in degrees we can use the DEGREES function:

which returns approx. 53.13°

We can also calculate the ratio based on the angle value using the SIN function:

This formula returns 0.8 which matches the ratio between the opposite side (4) and the hypotenuse (5) which is 4/5=0.8

The image above shows a right-angled triangle in blue, the opposite side named a is equal to 4. The adjacent side named b is equal to 3, the hypotenuse named c is equal to 5. A right-angled triangle means that one of the internal angles is equal to π/2 radians (90°).

**C** = π/2 radians (90°)

**A** = 0.927 radians (53.13°)

**B** = 180° - 90° - 53.13° = 36.87°

## 4. Example 2

**Calculate the angle (in radians) between the horizontal and the line joining the points (0, 0) and (4, 3) in the Cartesian plane, use the ASIN function?**

The Cartesian coordinate system specifies each point by a pair of real numbers called coordinates x and y (x,y). The question describes a line from (0,0) to (4,3), this means that x is equal to 4 and y is equal to 3.

This tells us that the opposite side in the triangle is 3 (a) and the adjacent side is 4 (b).

To calculate the hypotenuse we can use Pythagoras theorem which states that the squared hypotenuse is equal to the sum of the squared opposite side and the adjacent side. c^{2} = a^{2} + b^{2}

c = (3^{2}+4^{2})^{1/2} = (9+16)^{1/2} = 25^{1/2} = 5.

The argument is:

- number = opposite / hypotenuse = a/c = 3/5 = 0.6

Formula in cell C20:

The formula in cell C20 returns 0.643501108793284 radians which represents the angle between the line (0,0) - (4,3) and the horizontal dashed black line, in the image above. To get the result in degrees we can use the DEGREES function:

which returns approx. 36.87°

We can also calculate the ratio based on the angle using the SIN function:

This formula returns 0.6 which matches the ratio between the opposite side (3) and the hypotenuse (5) which is 3/5=0.6

### Functions in 'Math and trigonometry' category

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

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