For the inverse trigonometric function of sine 1/2 we usually employ the abbreviation arcsin and write it as arcsin 1/2 or arcsin(1/2).

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If you have been looking for what is arcsin 1/2, either in degrees or radians, or if you have been wondering about the inverse of sin 1/2, then you are right here, too.

In this post you can find the angle arcsine of 1/2, along with identities.

Arcsin of 1/2

If you want to know what is arcsin 1/2 in terms of trigonometry, check out the explanations in the last paragraph; ahead in this section is the value of arcsine(1/2):

arcsin 1/2 = π/6 rad = 30°arcsine 1/2 = π/6 rad = 30 °arcsine of 1/2 = π/6 radians = 30 degrees" onclick="if (!window.__cfRLUnblockHandlers) return false; return fbs_click()" target="_blank" rel="nofollow noopener noreferrer" data-cf-modified-ee827e69aafa422f1e04b055-="">

The arcsin of 1/2 is π/6 radians, and the value in degrees is 30°. To change the result from the unit radian to the unit degree multiply the angle by 180° / $\pi$ and obtain 30°.

Our results above contain fractions of π for the results in radian, and are exact values otherwise. If you compute arcsin(1/2), and any other angle, using the calculator below, then the value will be rounded to ten decimal places.

To obtain the angle in degrees insert 1/2 as decimal in the field labelled “x”. However, if you want to be given the angle opposite to 1/2 in radians, then you must press the swap units button.

Calculate arcsin x

x:
A Really Cool Arcsine Calculator and Useful Information! Please ReTweet. Click To TweetA Really Cool Arcsine Calculator and Useful Information! Please ReTweet. Click To TweetApart from the inverse of sin 1/2, similar trigonometric calculations include:

The identities of arcsine 1/2 are as follows: arcsin(1/2) =

$\frac{\pi}{2}$ – arcscos(1/2) ⇔ 90°- arcscos(1/2) -arcsin(-1/2) arccsc(1/1/2) $\frac{arccos(1-2(1/2)^{2})}{2}$ $2 arctan(\frac{1/2}{1 + \sqrt{1 – (1/2)^{2}}})$

The infinite series of arcsin 1/2 is: $\sum_{n=0}^{\infty} \frac{(2n)!}{2^{2n}(n!)^{2}(2n+1)}(1/2)^{2n+1}$.

Next, we discuss the derivative of arcsin x for x = 1/2. In the following paragraph you can additionally learn what the search calculations form in the sidebar is used for.

Derivative of arcsin 1/2

The derivative of arcsin 1/2 is particularly useful to calculate the inverse sine 1/2 as an integral.

The formula for x is (arcsin x)’ = $\frac{1}{\sqrt{1-x^{2}}}$, x ≠ -1,1, so for x = 1/2 the derivative equals 1.1547005384.

Using the arcsin 1/2 derivative, we can calculate the angle as a definite integral:

arcsin 1/2 = $\int_{0}^{1/2}\frac{1}{\sqrt{1-z^{2}}}dz$.

The relationship of arcsin of 1/2 and the trigonometric functions sin, cos and tan is:

sin(arcsine(1/2)) = 1/2 cos(arcsine(1/2)) = $\sqrt{1 – (1/2)^{2}}$ tan(arcsine(1/2)) = $\frac{1/2}{\sqrt{1 – (1/2)^{2}}}$

Note that you can locate many terms including the arcsine(1/2) value using the search form. On mobile devices you can find it by scrolling down. Enter, for instance, arcsin1/2 angle.

Using the aforementioned form in the same way, you can also look up terms including derivative of inverse sine 1/2, inverse sine 1/2, and derivative of arcsin 1/2, just to name a few.

In the next part of this article we discuss the trigonometric significance of arcsine 1/2, and there we also explain the difference between the inverse and the reciprocal of sin 1/2.

What is arcsin 1/2?

In a triangle which has one angle of 90 degrees, the sine of the angle α is the ratio of the length of the opposite side o to the length of the hypotenuse h: sin α = o/h.

In a circle with the radius r, the horizontal axis x, and the vertical axis y, α is the angle formed by the two sides x and r; r moving counterclockwise defines the positive angle.

As follows from the unit-circle definition on our homepage, assumed r = 1, in the intersection of the point (x,y) and the circle, y = sin α = 1/2 / r = 1/2. The angle whose sine value equals 1/2 is α.

In the interval <-π/2, π/2> or <-90°, 90°>, there is only one α whose sine value equals 1/2. For that interval we define the function which determines the value of α as y = arcsin(1/2)." onclick="if (!window.__cfRLUnblockHandlers) return false; return fbs_click()" target="_blank" rel="nofollow noopener noreferrer" data-cf-modified-ee827e69aafa422f1e04b055-="">

From the definition of arcsin(1/2) follows that the inverse function y-1 = sin(y) = 1/2. Observe that the reciprocal function of sin(y),(sin(y))-1 is 1/sin(y).

Avoid misconceptions and remember (sin(y))-1 = 1/sin(y) ≠ sin-1(y) = arcsin(1/2). And make sure to understand that the trigonometric function y=arcsine(x) is defined on a restricted domain, where it evaluates to a single value only, called the principal value:

In order to be injective, also known as one-to-one function, y = arcsine(x) if and only if sin y = x and -π/2 ≤ y ≤ π/2. The domain of x is −1 ≤ x ≤ 1.

Conclusion

The frequently asked questions in the context include what is arcsin 1/2 degrees and what is the inverse sine 1/2 for example; reading our content they are no-brainers.

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