Arcsine, written as arcsin or sin-1 (not khổng lồ be confused with ), is the inverse sine function. Sine only has an inverse on a restricted domain, ≤x≤. In the figure below, the portion of the graph highlighted in red shows the portion of the graph of sin(x) that has an inverse.

Bạn đang xem: Arcsin The tên miền must be restricted because in order for a function lớn have an inverse, the function must be one-to-one, meaning that no horizontal line can intersect the graph of the function more than once. Since sine is a periodic function, without restricting the domain, a horizontal line would intersect the function periodically, infinitely many times.

One of the properties of inverse functions is that if a point (a, b) is on the graph of f, the point (b, a) is on the graph of its inverse. This effectively means that the graph of the inverse function is a reflection of the graph of the function across the line y = x.

The graph of y = arcsin(x) is shown below. As can be seen from the figure, y = arcsin(x) is a reflection of sin(x), given the restricted domain ≤x≤, across the line y = x. The tên miền of arcsin(x), -1≤x≤1, is the range of sin(x), and its range, ≤y≤, is the domain name of sin(x).

## Arcsine calculator

The following is a calculator lớn find out either the arccos value of a number between -1 và 1 or cosine value of an angle.

## Using special angles lớn find arcsin

While we can find the value of arcsine for any x value in the interval <-1, 1>, there are certain angles that are used frequently in trigonometry (0°, 30°, 45°, 60°, 90°, và their multiples và radian equivalents) whose sine & arcsine values may be worth memorizing. Below is a table showing these angles (θ) in both radians & degrees, and their respective sine values, sin(θ).

 θ -90° -60° -45° -30° 0° 30° 45° 60° 90° sin(θ) -1 0 1

One method that may help with memorizing these values is lớn express all the values of sin(θ) as fractions involving a square root. Starting from 0° progressing through 90°, sin(0°) = 0 = . The subsequent values, sin(30°), sin(45°), sin(60°), and sin(90°) follow a pattern such that using the value of sin(0°) as a reference, khổng lồ find the values of sine for the subsequent angles, we simply increase the number under the radical sign in the numerator by 1, as shown below.

 θ 0° 30° 45° 60° 90° sin(θ)   The values of sine from 0° through -90° follows the same pattern except that the values are negative instead of positive since sine is negative in quadrant IV. This pattern repeats periodically for the respective angle measurements, and we can identify the values of sin(θ) based on the position of θ in the unit circle, taking the sign of sine into consideration: sine is positive in quadrants I & II and negative in quadrants III and IV.

Once we"ve memorized the values, or if we have a reference of some sort, it becomes relatively simple lớn recognize & determine sine or arcsine values for the special angles.

Example:

Find arcsin(), arcsin(), & arcsin(3) in radians. , . , .

arcsin(3) is undefined because 3 is not within the interval -1≤arcsin(θ)≤1, the tên miền of arcsin(x).

## Inverse properties

Generally, functions and their inverses exhibit the relationship

f(f-1(x)) = x và f-1(f(x)) = x

given that x is in the tên miền of the function. The same is true of sin(x) và arcsin(x) within their respective restricted domains:

sin(arcsin(x)) = x, for all x in <-1, 1>

and

arcsin(sin(x)) = x, for all x in <, >

These properties allow us to lớn evaluate the composition of trigonometric functions.

### Composition of arcsine and sine

If x is within the domain, evaluating a composition of arcsine and sine is relatively simple.

Examples:

1. 2. If x is not within the domain, we need khổng lồ determine the reference angle as well as the relevant quadrant. Given arcsin(sin() ), we cannot evaluate this as we did above because x is not within <, >, so the solution cannot be . To evaluate this, we first need khổng lồ determine sin() before using arcsin:

3. In the above example, the reference angle is and sin() is . However, is in quadrant III where sin is negative, so sin() = , và the only angle within the domain of arcsin(x) whose sine is is .

### Composition of other trigonometric functions

We can also make compositions using all the other trigonometric functions: cosine, tangent, cosecant, secant, & cotangent.

Example:

Find cos(arcsin()).

Since is not one of the ratios for the special angles, we can use a right triangle khổng lồ find the value of this composition. Given arcsin()=θ, we can find that sin(θ)=. The right triangle below shows θ & the ratio of its opposite side lớn the triangle"s hypotenuse. To find cosine, we need khổng lồ find the adjacent side since cos(θ)=. Let b be the length of the adjacent side. Using the Pythagorean Theorem,

32 + b2 = 52

9 + b2 = 25

b2 = 16

b = 4

We know that arcsin() = θ, so we can rewrite the problem and find cos(θ) by using the triangle we constructed above & the fact that cos(θ)=:

cos(arcsin()) = cos(θ) = Example:

Find tan(arcsin(2x)).

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Given arcsin(2x) = θ, we can find that sin(θ) = và construct the following triangle: To find tangent, we need to lớn find the adjacent side since tan(θ)= . Let b be the length of the adjacent side. Using the Pythagorean theorem,

(2x)2 + b2 = 12

4x2 + b2 = 1

b2 = 1 - 4x2

b = and

tan(arcsin(2x)) = tan(θ) = , where

2. 2sin2(x) + 5sin(x) - 3 = 0

2sin2(x) + 5sin(x) - 3 = 0

(2sin(x) - 1)(sin(x) + 3) = 0

2sin(x) - 1 = 0 or sin(x) + 3 = 0

sin(x) = or sin (x) = -3

x = arcsin() or x = arcsin(-3)

Solving for x = arcsin(),

x= or

We cannot solve for x = arcsin(-3) because it is undefined, so x= or are the only solutions.