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Arccos 1/2

Welcome to arccos 1/2, our post aboutthe arccosine of 1/2.

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

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

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

Read on to learn all about the arccos of 1/2.

Arccos of 1/2

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

arccos 1/2 = π/3 rad = 60°
arccosine 1/2 = π/3 rad = 60 °
arccosine of 1/2 = π/3 radians = 60 degrees

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The arccos of 1/2 is π/3 radians, and the value in degrees is 60°. To change the result from the unit radian to the unit degree multiply the angle by 180° / $\pi$ and obtain 60°.

Our results above contain fractions of π for the results in radian, and are exact values otherwise. If you compute arccos(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 adjacent to 1/2 in radians, then you must press the swap units button.

Calculate arccos x

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

The identities of arccosine 1/2 are as follows: arccos(1/2) =

  • $\frac{\pi}{2}$ – arcsin(1/2) ⇔ 90°- arcsin(1/2)
  • ${\pi}$ – arccos(-1/2) ⇔ 180° – arcos(-1/2)
  • arcsec(1/1/2)
  • $arcsin(\sqrt{1-(1/2)^{2}})$
  • $2arctan(\frac{\sqrt{1-(1/2)^{2}}}{1+(1/2)})$

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

Next, we discuss the derivative of arccos 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 arccos 1/2

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

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

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

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

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

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

Note that you can locate many terms including the arccosine(1/2) value using the search form. On mobile devices you can find it by scrolling down. Enter, for instance, arccos1/2 angle.

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

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

What is arccos 1/2?

In a triangle which has one angle of 90 degrees, the cosine of the angle α is the ratio of the length of the adjacent side a to the length of the hypotenuse h: cos α = a/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, x = cos α = 1/2 / r = 1/2. The angle whose cosine value equals 1/2 is α.

In the interval [0, π] or [0°, 180°], there is only one α whose arccosine value equals 1/2. For that interval we define the function which determines the value of α as

y = arccos(1/2).

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

Avoid misconceptions and remember (cos(y))-1 = 1/cos(y) ≠ cos-1(y) = arccos(1/2). And make sure to understand that the trigonometric function y=arccos(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 = arccos(x) if and only if cos y = x and 0 ≤ y ≤ π. The domain of x is −1 ≤ x ≤ 1.

Conclusion

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

But, if there is something else about the topic you would like to know, fill in the form on the bottom of this post, or send us an email with a subject line such as arccosine 1/2 in radians.

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

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