Domain And Range Of Sine

Arcsin

Arcsin is ane of the six main inverse trigonometric functions. Information technology is the inverse trigonometric function of the sine function. Arcsin is also chosen inverse sine and is mathematically written as arcsin ten or sin^-aneten (read equally sine inverse 10). An of import thing to note is that sin^-1x is not the same as (sin 10)^-1, that is, sin^-1x is non the reciprocal function of sin x. In inverse trigonometry, we have vi inverse trigonometric functions - arccos, arcsin, arctan, arcsec, arccsc, and arccot.

Arcsin x gives the measure of the angle corresponding to the ratio of the perpendicular and hypotenuse of a right-angled triangle. In this article, we will explore the concept of arcsin and derive its formula. We will as well discuss the domain and range of arcsin x and hence, plot its graph. We will also solve diverse examples using the identities of arcsin x to understand its applications and the concept better.

one.	What is Arcsin?
2.	Arcsin x Formula
3.	Arcsin Graph
4.	Domain and Range of Arcsin
5.	Arcsin Identities
6.	FAQs on Arcsin

What is Arcsin?

Arcsin is the changed trigonometric office of the sine function. It gives the measure of the angle for the corresponding value of the sine function. We denote the arcsin part for the real number x equally arcsin x (read as arcsine x) or sin^-onex (read every bit sine inverse ten) which is the inverse of sin y. If sin y = x, so nosotros can write it as y = arcsin x. Arcsin is one of the six important changed trigonometric functions. The 6 inverse trigonometric functions are:

Arcsin: Inverse of sine function, denoted by arcsin x or sin^-1x
Arccos: Inverse of cosine role, denoted by arccos x or cos^-110
Arctan: Inverse of tangent role, denoted by arctan x or tan^-110
Arccot: Inverse of cotangent function, denoted by arccot x or cot^-ix
Arcsec: Inverse of secant function, denoted past arcsec x or sec^-110
Arccsc: Changed of cosecant office, denoted by arccsc 10 or csc^-1x

The arcsin function helps us detect the mensurate of an angle respective to the sine role value. Let usa see a few examples to sympathise its operation. We know the values of the sine function for some specific angles using the trigonometric table.

If sin 0 = 0, and so arcsin 0 = 0
sin π/6 = 1/ii implies arcsin (i/2) = π/vi
sin π/3 = √3/2 implies arcsin (√3/2) = π/3
If sin π/2 = i, then arcsin (1) = π/2

Arcsin x Formula

Nosotros can employ the arcsin formula when the value of sine of an angle is given and nosotros want to evaluate the exact measure of the bending. Consider a correct-angled triangle. Nosotros know that sin θ = Opposite Side / Hypotenuse. As arcsin is the inverse office of the sine office, therefore, we accept θ = arcsin (Opposite Side / Hypotenuse). Therefore, the formula for arcsin x is,

θ = arcsin (Opposite Side / Hypotenuse)

Arcsin Formula

We can also apply the police of sines to derive the arcsin formula. For a triangle ABC with sides AB = c, BC = a and AC = b, nosotros take sin A / a = sin B / b = sin C / c. Then, taking two at a time, we take

sin A / a = sin B / b

⇒ sin A = (a/b) sin B

⇒ A = arcsin [(a/b) sin B]

Similarly, we can find the measure out of the angles B and C using the same method.

Arcsin Graph

Now that we know the arcsin formula, we will plot the graph of arcsin x using some of its points. As discussed the functioning of arcsin, nosotros know the values of the sine office for some specific angles and using trigonometric formulas, nosotros have

sin 0 = 0 implies arcsin 0 = 0 → (0, 0)
sin π/6 = 1/2 implies arcsin (i/two) = π/6 → (1/2, π/vi)
sin π/iii = √iii/2 implies arcsin (√iii/2) = π/3 → (√iii/2, π/3)
sin π/2 = 1 implies arcsin (1) = π/2 → (1, π/two)
sin (-π/4) = -i/√2 implies arcsin (-1/√ii) = -π/iv → (-1/√two, -π/iv)
sin (-π/6) = -1/2 implies arcsin (-1/two) = -π/6 → (-1/2, -π/six)

Now, by plotting the above points on a graph, nosotros have the graph of arcsin given beneath:

Arcsin Graph

Domain and Range of Arcsin

As we know that two functions are inverses of each other if they are bijective and the domain and range of the office get the range and domain, respectively of the inverse function. We know that the domain of sin x is all real numbers and its range is [-ane, one]. Only with this domain, sin x is not bijective. So, nosotros restrict the domain of sine function to [–π/2, π/2], and then sin x becomes bijective with domain [–π/2, π/2] and range [-1, 1]. When the domain of sin x is restricted to [–3π/two, –π/2], [–π/2, π/2], or [π/2, 3π/2], and and then on, and range [-1, 1], so sin x is bijective and hence, correspondingly we can define arcsin with domain [-1, 1] and range [–3π/2, –π/2], [–π/2, π/2], or [π/2, 3π/2], then on.

We get different branches of the arcsin role for each interval. The branch of arcsin corresponding to domain [-ane, 1] and range [–π/2, π/2] is called the chief value branch. So, the arcsin is defined every bit arcsin: [-1, i] → [–π/2, π/2]. Hence, the domain and range of arcsin are:

Domain of Arcsin: [-ane, ane]
Range of Arcsin: [–π/2, π/2]

Arcsin Identities

Now, nosotros volition discuss some of the of import properties and identities of the arcsin function that help usa to simplify and solve diverse problems in trigonometry.

sin (arcsin ten) = x, if ten is in [-1, 1]
arcsin (sin ten) = ten, if 10 is in [–π/2, π/2]
arcsin (1/x) = arccsc ten, if x ≤ -1 or x ≥ 1
arcsin (–x) = - arcsin 10, if x ∈ [-one, ane]
arcsin ten + arccos 10 = π/2, if ten ∈ [-1, 1]
2 arcsin x = arcsin (2x √(1 - x²)), if -one/√2 ≤ 10 ≤ i/√2
2 arccos 10 = arcsin (2x √(i - x^two)), if 1/√ii ≤ x ≤ ane
arcsin x + arcsin y = arcsin [x√(one - y²) + y√(one - ten^two)]

Important Notes on Arcsin

Arcsin is the inverse function of sine part.
The domain and range of arcsin are [-i, 1] and [–π/two, π/two], respectively.
The derivative of arcsin is i/√(1 - x²).
The integral of arcsin is ∫arcsin x dx = x sin^-1ten + √(1 - x^two) + C

☛ Related Topics:

Sin 1 in Degrees
Changed Trigonometric Ratios
Changed Trig Derivatives

Arcsin Examples

Instance i: Show the arcsin formula 2 arcsin x = arcsin (2x √(1 - x^two)), if -1/√2 ≤ ten ≤ 1/√2.

Solution: Assume arcsin x = y, then nosotros have sin y = ten. Consider RHS

RHS = arcsin (2x √(1 - x^two))

= arcsin [2 sin y √(ane - sin^twoy)]

= arcsin [ii sin y √(cos²y)] --- [Using trigonometric formula sin²A + cos²A = i which implies cos²A = 1 - sin²A]

= arcsin [two sin y cos y]

= arcsin [sin2y] --- [Using trigonometric formula sin2A = 2 sinA cosA]

= 2y

= two arcsin x --- [Because arcsin 10 = y]

Answer: Hence, we take proved 2 arcsin x = arcsin (2x √(i - x²)), if -1/√2 ≤ x ≤ 1/√2
Example two: Find the value of arcsin (sin 3π/5).

Solution: Nosotros know that arcsin (sin x) = x, so nosotros take arcsin (sin 3π/five) = 3π/5 but 3π/v ∉ [–π/2, π/ii]. So, we demand to detect the value equivalent to sin 3π/five such that the angle lies in the interval [–π/two, π/ii]. Using trigonometric formula sin x = sin (π - x), we have

sin (3π/5) = sin (π - 3π/v)

= sin (5π/5 - 3π/5)

= sin (2π/v)

As well, note that 2π/5 ∈ [–π/2, π/2].

And then, nosotros have arcsin (sin 3π/5) = 2π/5

Answer: arcsin (sin 3π/5) = 2π/five
Instance three: Evidence that arcsin (3/5) – arcsin (8/17) = arccos (84/85)

Solution: Assume A = arcsin (3/5) and B = arcsin (8/17), then we have sin A = 3/5 and sin B = 8/17. Then using trigonometric formula, sin²x + cos²x = i, we have

cos A = √ (ane - sin²A)

= √ (1 - (3/5)²)

= √(one - 9/25)

= √(16/25)

= 4/five

cos B = √ (1 - sin²B)

= √ (1 - (8/17)²)

= √(1 - 64/289)

= √(225/289)

=xv/17

Now, using the formula cos (A - B) = cos A cos B + sin A sin B

= 4/5 × fifteen/17 + three/v × viii/17

= lx/85 + 24/85

= 84/85

⇒ A - B = arccos (84/85)

⇒ arcsin (3/5) – arcsin (8/17) = arccos (84/85) --- [A = arcsin (3/five) and B = arcsin (8/17)]

Answer: Hence, we take proved that arcsin (three/v) – arcsin (8/17) = arccos (84/85)

go to slidego to slidego to slide

Breakup tough concepts through uncomplicated visuals.

Math will no longer be a tough subject area, particularly when y'all understand the concepts through visualizations.

Book a Gratuitous Trial Class

Exercise Questions on Arcsin Questions

go to slidego to slide

FAQs on Arcsin

What is Arcsin in Trigonometry?

Arcsin is an inverse trigonometric function of the sine part. Nosotros denote the arcsin function for the real number x every bit arcsin x (read equally arcsine x) or sin^-anex (read as sine inverse 10). It is i of the half dozen master inverse trigonometric functions give by, arccos, arcsin, arctan, arcsec, arccsc, and arccot. An important thing to keep in mind is that sin^-anex is non the reciprocal of sine.

What is Arcsin Formula?

The formula for arcsin is given past, θ = arcsin (Opposite Side / Hypotenuse), where θ is the angle in a right-angled triangle. The arcsin function helps us observe the measure out of an angle corresponding to the sine role value. We can besides find the measure out of an bending in a triangle using the arcsin formula derived using the law of sines.

What is the Derivative of Arcsin x?

The derivative of arcsin is given by, d/dx(arcsin x) = one/√(i - x²). Nosotros tin derive this formula using the first principle of derivatives and the concatenation dominion method of differentiation.

How to Integrate Arcsin?

The integral of arcsin is given by, ∫arcsin x dx = x sin^-1x + √(1 - x²) + C, where C is the abiding of integration. It can be derived using unlike methods such every bit integration by parts and substitution method followed by integration by parts.

What is the Domain and Range of Arcsin?

The domain and range of arcsin are:

Domain of Arcsin: [-1, i]
Range of Arcsin: [–π/2, π/2]

We restrict the domain of the sine function to [–π/two, π/two] to make it bijective and hence, define the arcsin office as two functions are inverses of each other if they are bijective. The branch of arcsin corresponding to domain [-ane, 1] and range [–π/ii, π/two] is called the master value co-operative.

How to Plot the Arcsin Graph?

Using the definition and functioning of arcsin, nosotros can plot some points on the graph with the help of a trigonometric tabular array. Some of the points are:

sin 0 = 0 implies arcsin 0 = 0 → (0, 0)
sin π/six = 1/2 implies arcsin (ane/ii) = π/vi → (ane/two, π/vi)
sin π/three = √3/2 implies arcsin (√3/two) = π/three → (√3/two, π/3)
sin π/2 = 1 implies arcsin (1) = π/2 → (i, π/2)
sin (-π/4) = -1/√2 implies arcsin (-ane/√2) = -π/four → (-1/√2, -π/4)
sin (-π/6) = -1/two implies arcsin (-1/2) = -π/6 → (-ane/ii, -π/6)

Then, by plotting these points and joining through a bend, we get the arcsin graph.

Is Arcsin the Inverse of Sin?

Arcsin is the inverse of the trigonometric function sin. When the arcsin function is defined as arcsin: [-one, 1] → [–π/2, π/ii], and then we say that it is the inverse of sin: [–π/ii, π/2] → [-1, 1].

What is the Difference between Sin and Arcsin?

Sine is a trigonometric function that maps a real number to an bending whereas arcsin is the inverse of the sine role. Both functions are divers as arcsin: [-1, 1] → [–π/2, π/2], so we say that it is the inverse of sin: [–π/two, π/2] → [-ane, 1] and are inverses of each other.

Why Arcsin (-ii) is Non Defined?

Arcsin (-ii) is non defined because the domain of arcsin is restricted to [-1, one] and -2 does not prevarication in the interval [-1, 1].

What are the Identities of Arcsin?

Some of the important formulas and identities of arcsin are:

sin (arcsin ten) = x, if x is in [-i, 1]
arcsin (sin ten) = x, if x is in [–π/2, π/2]
arcsin (1/x) = arccsc x, if x ≤ -1 or x ≥ 1
arcsin (–ten) = - arcsin x, if x ∈ [-1, 1]
arcsin 10 + arccos ten = π/2, if x ∈ [-1, ane]
2 arcsin x = arcsin (2x √(i - x²)), if -i/√ii ≤ ten ≤ ane/√ii

What is Arcsin of Sin?

The formula for arcsin of sin is given by, arcsin (sin ten) = ten, if x is in [–π/ii, π/two].