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Arcsin sqrt(2)/2

Welcome to arcsin sqrt(2)/2, our post aboutthe arcsine of sqrt(2)/2.

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

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

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

Read on to learn all about the arcsin of sqrt(2)/2, and note that the term sqrt(2)/2 is approximately 0.707106781 as a decimal number.

Arcsin of sqrt(2)/2

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

arcsin sqrt(2)/2 = pi/4 rad = 45°
arcsine sqrt(2)/2 = pi/4 rad = 45 °
arcsine of sqrt(2)/2 = pi/4 radians = 45 degrees

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The arcsin of sqrt(2)/2 is pi/4 radians, and the value in degrees is 45°. To change the result from the unit radian to the unit degree multiply the angle by 180° / $\pi$ and obtain 45°.

Our results above contain fractions of pi for the results in radian, and are exact values otherwise. If you compute arcsin(sqrt(2)/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 sqrt(2)/2 as decimal in the field labelled “x”. However, if you want to be given the angle opposite to sqrt(2)/2 in radians, then you must press the swap units button.

Calculate arcsin x

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

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

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

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

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

Derivative of arcsin sqrt(2)/2

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

The formula for x is (arcsin x)’ = $\frac{1}{\sqrt{1-x^{2}}}$, x ≠ -1,1, so for x = sqrt(2)/2 the derivative equals 1.4142135624.

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

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

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

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

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

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

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

What is arcsin sqrt(2)/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 = sqrt(2)/2 / r = sqrt(2)/2. The angle whose sine value equals sqrt(2)/2 is α.

In the interval [-pi/2, pi/2] or [-90°, 90°], there is only one α whose sine value equals sqrt(2)/2. For that interval we define the function which determines the value of α as

y = arcsin(sqrt(2)/2).

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From the definition of arcsin(sqrt(2)/2) follows that the inverse function y-1 = sin(y) = sqrt(2)/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(sqrt(2)/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 -pi/2 ≤ y ≤ pi/2. The domain of x is −1 ≤ x ≤ 1.

Conclusion

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

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– Article written by Mark, last updated on February 4th, 2017

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