Arcsin 1 (Sin Inverse 1)

The value of arcsin 1 is equal to 90 degrees or π/2 radians. Before we learn how to evaluate the value of sin inverse 1, let us first recall the meaning of the inverse trigonometric function arcsin. Arcsin or Sin inverse is the inverse function of the trigonometric function sine, denoted by arcsin x or sin^-1x. Arcsin 1 or sin inverse 1 gives the value of the angle corresponding to the value when the sine function is equal to 1. We can find the value of sin inverse 1 using the unit circle and its coordinates which are given by (cos x, sin x).

Further, in this article, we will determine the value of arcsin 1 using a unit circle and trigonometry table in degrees and radians (the two units of measure of an angle). We will also express the value of sin inverse 1 in terms of pi (π). We shall solve a few examples and evaluate the value of the arcsin function for different values for a better understanding of the concept.

Arcsin 1 gives the measure of the angle for which the value of the sine function is equal to 1. Sine and Arcsin are inverses of each other defined by, sin: [–π/2, π/2] → [-1, 1] and arcsin: [-1, 1] → [–π/2, π/2] such that if sin x = y, then x = sin^-1y. Using the trigonometry table, we know that sin (π/2) = 1 or sin 90° = 1. So, using the definition of sin inverse, we have the value of arcsin 1 as sin^-1(1) = π/2 rad or 90°.

We know that the coordinates of the points on a unit circle are given by (cos θ, sin θ), where θ is the angle made with the positive x-axis, x-coordinate is cos θ and y-coordinate is sin θ. To find the value of sin inverse 1, we need to find the angle made with the positive x-axis when the y-coordinate, sin θ is equal to 1. Given below is an image of a unit circle (a circle with a radius equal to 1 unit). As we can see in the image, when the angle made with the positive x-axis (in anticlockwise direction) is equal to π/2 radians, then the coordinates of the point on the circle are (0, 1).

So, we have (cos θ, sin θ) = (0, 1) when the angle with the positive x-axis in anticlockwise direction is equal to π/2 radians or 90 degrees. Therefore, the value of sin θ is equal to 1, when the angle is π/2 radians. In other words, we can say that θ is equal to π/2 radians when sin θ is equal to 1 which implies the value of sin inverse 1 is equal to π/2 rad or 90 degrees.

Now that we know the value of arcsin 1, let us express it in degrees. As discussed before, the arcsin function is defined as arcsin: [-1, 1] → [–90°, 90°], that is, its domain and range are [-1, 1] and [-90°, 90°], respectively. Also, as we know that sine function is periodic, we know that sin θ = 1 for infinitely many values of θ = 90°, 450°, 810°, and so on. But since we have restricted the domain of the sine function to [–90°, 90°], we have sine function is equal to 1 only when the angle is 90° which lies in the principal value interval [–90°, 90°]. So, the value of arcsin 1 in degrees is equal to 90 degrees, that is, arcsin 1 = 90°.

As we know that the principal branch of arcsin is defined as arcsin: [-1, 1] → [–π/2, π/2], where [–π/2, π/2] is the range of sin inverse function. So, using the trigonometry table, we have

arcsin 1 = θ

⇒ sin θ = 1

⇒ sin θ = sin π/2 --- [Because sin π/2 = 1]

⇒ θ = π/2, where π/2 lies in the principal branch range [–π/2, π/2]

Therefore, the value of sin inverse 1 is equal to π/2 radians. This can also be interpreted as arcsin 1 in radians is equal to 1.57079633 rad.

As we know that arcsin 1 is equal to 90 degrees, we convert this value of sin inverse 1 in radians to express it in pi using the degrees to radians formula. The formula to convert degrees to radians is given by,

Angle in Radians = Angle in Degrees × π / 180° --- [Multiplying the angle in degrees by π/180]

⇒ Arcsin 1 in terms of π = 90° × π / 180°

= π/2 radians

Hence, arcsin 1 in terms of pi is given by π/2 rad.

Important Notes on Arcsin 1

• Arcsin 1 is equal to 90° or π/2 radians.
• Arcsin function is defined by arcsin from [-1, 1] → [–90°, 90°].
• Sin inverse 1 in radians is equal to 1.57079633 rad.

☛ Related Topics:

• Derivative of Arcsin
• Integral of Sin inverse
• Arccosine

What is Arcsin 1?

Arcsin 1 gives the measure of the angle for which the value of the sine function is equal to 1. The value of arcsin 1 is equal to 90 degrees or π/2 radians.

How to Find The Value of Sin Inverse 1?

We can determine the value of arcsin 1 using a unit circle and trigonometry table in degrees and radians. We know that the coordinates of the points on a unit circle are given by (cos θ, sin θ). To find the value of sin inverse 1, we need to find the angle made with the positive x-axis when the y-coordinate, sin θ is equal to 1.

What is Sin Inverse 1 in Degrees?

The value of sin inverse 1 in degrees is equal to 90 degrees, that is, arcsin 1 = 90°.

What is Arcsin 1 in Radians?

The value of sin inverse 1 is equal to π/2 radians. This can also be interpreted as arcsin 1 in radians is equal to 1.57079633 rad.

How to Write the Value of Arcssin 1 in Terms of Pi?

Sin inverse 1 in terms of pi is given by π/2 rad. We have

sin θ = sin π/2 --- [Because sin π/2 = 1]

⇒ θ = π/2

What is the Value of Arcsin 1 by Root 2?

The value of arcsin 1 by root 2 is 45 degrees because sin 45° = 1/√2.

What is Sin Inverse 1 By 2?

The value of sin inverse 1 by 2 is equal to π/6 radians because sin (π/6) = 1/2 and π/6 lies in the interval [–π/2, π/2].

What is the Value of Arcsin (-1)?

The value of arcsin (-1) is equal to -π/2 rad because arcsin (-1) = - arcsin 1 = - π/2 rad.