Why Is Arcsin 2 Undefined

Arcsin

Arcsine, written as arcsin or sin^-i (not to be confused with ), is the inverse sine function. Sine only has an inverse on a restricted domain, ≤10≤. In the effigy below, the portion of the graph highlighted in red shows the portion of the graph of sin(10) that has an inverse.

The domain must be restricted because in order for a role to take an changed, the office must be one-to-one, meaning that no horizontal line tin intersect the graph of the function more than than once. Since sine is a periodic office, without restricting the domain, a horizontal line would intersect the function periodically, infinitely many times.

I of the properties of inverse functions is that if a signal (a, b) is on the graph of f, the signal (b, a) is on the graph of its inverse. This effectively ways that the graph of the inverse function is a reflection of the graph of the part across the line y = x.

The graph of y = arcsin(x) is shown beneath.

As can be seen from the effigy, y = arcsin(ten) is a reflection of sin(x), given the restricted domain ≤10≤, across the line y = x. The domain of arcsin(x), -1≤x≤1, is the range of sin(x), and its range, ≤y≤, is the domain of sin(x).

Arcsine calculator

The following is a calculator to find out either the arccos value of a number betwixt -1 and 1 or cosine value of an angle.

Using special angles to find arcsin

While nosotros can detect the value of arcsine for any x value in the interval [-1, 1], in that location are certain angles that are used frequently in trigonometry (0°, 30°, 45°, 60°, 90°, and their multiples and radian equivalents) whose sine and arcsine values may be worth memorizing. Beneath is a table showing these angles (θ) in both radians and degrees, and their corresponding sine values, sin(θ).

θ	-xc°	-60°	-45°	-30°	0°	thirty°	45°	60°	xc°
sin(θ)	-1				0				1

One method that may help with memorizing these values is to limited all the values of sin(θ) every bit fractions involving a foursquare root. Starting from 0° progressing through ninety°, 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°) equally a reference, to observe the values of sine for the subsequent angles, we simply increment the number under the radical sign in the numerator past 1, as shown beneath.

θ	0°	xxx°	45°	threescore°	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 Four. This pattern repeats periodically for the corresponding angle measurements, and we tin 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 and Ii and negative in quadrants Three and IV.

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

Example:

Find arcsin(), arcsin(), and arcsin(three) in radians.

, .

arcsin(3) is undefined because 3 is not within the interval -1≤arcsin(θ)≤1, the domain of arcsin(ten).

Inverse properties

Generally, functions and their inverses showroom the relationship

f(f^-1(x)) = 10 and f^-1(f(x)) = x

given that x is in the domain of the function. The aforementioned is true of sin(10) and arcsin(x) inside their respective restricted domains:

sin(arcsin(x)) = x, for all x in [-one, 1]

and

arcsin(sin(x)) = x, for all x in [, ]

These backdrop allow us to evaluate the composition of trigonometric functions.

Composition of arcsine and sine

If ten is within the domain, evaluating a composition of arcsine and sine is relatively elementary.

Limerick of other trigonometric functions

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

Example:

Notice cos(arcsin()).

Since is not one of the ratios for the special angles, nosotros can use a right triangle to discover the value of this composition. Given arcsin()=θ, nosotros can find that sin(θ)=. The correct triangle below shows θ and the ratio of its opposite side to the triangle's hypotenuse.

To find cosine, nosotros demand to find the adjacent side since cos(θ)=. Permit b be the length of the adjacent side. Using the Pythagorean Theorem,

iii² + b² = 5ⁱⁱ

nine + b^two = 25

bⁱⁱ = 16

b = 4

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

cos(arcsin()) = cos(θ) =

The same process can exist used with a variable expression.

Instance:

Find tan(arcsin(2x)).

Given arcsin(2x) = θ, we tin can detect that sin(θ) = and construct the following triangle:

To notice tangent, we need to find the adjacent side since tan(θ)=. Allow b be the length of the adjacent side. Using the Pythagorean theorem,

(2x)^two + bⁱⁱ = 1²

4x² + b² = one

b^two = 1 - 4x²

b =

and

tan(arcsin(2x)) = tan(θ) = , where <x<

Using arcsine to solve trigonometric equations

Arcsine tin also exist used to solve trigonometric equations involving the sine function.

Examples:

Solve the following trigonometric equations for x where 0≤ten<2π.

i. 2sin(10) =

2sin(ten) =

sin(x) =

Sine is positive in 2 quadrants, quadrant I and quadrant II, and so in that location are two solutions: 10= and x=. These are the just 2 angles inside 0≤x<2π whose sine value is equal to .

2. 2sin²(10) + 5sin(x) - iii = 0

2sin²(x) + 5sin(x) - iii = 0

(2sin(ten) - 1)(sin(x) + iii) = 0

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

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

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

Solving for x = arcsin(),

x= or

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