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Welcome to arctg sqrt(3)/3, our post aboutthe arctangent of sqrt(3)/3.
For the inverse trigonometric function of tangent sqrt(3)/3 we usually employ the abbreviation arctg and write it as arctg sqrt(3)/3 or arctg(sqrt(3)/3).
If you have been looking for what is arctg sqrt(3)/3, either in degrees or radians, or if you have been wondering about the inverse of tg sqrt(3)/3, then you are right here, too.
In this post you can find the angle arctangent of sqrt(3)/3, along with identities.
Read on to learn all about the arctg of sqrt(3)/3, and note that the term sqrt(3)/3 is approximately 0.577350269 as a decimal number.
Arctg of sqrt(3)/3
If you want to know what is arctg sqrt(3)/3 in terms of trigonometry, check out the explanations in the last paragraph; ahead in this section is the value of arctangent(sqrt(3)/3):
arctangent sqrt(3)/3 = π/6 rad = 30 °
arctangent of sqrt(3)/3 = π/6 radians = 30 degrees
The arctg of sqrt(3)/3 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 arctg(sqrt(3)/3), 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(3)/3 as decimal in the field labelled “x”. However, if you want to be given the angle of tg sqrt(3)/3 in radians, then you must press the swap units button.
Calculate arctg x
The identities of arctangent sqrt(3)/3 are as follows: arctg(sqrt(3)/3) =
- $\frac{\pi}{2}$ – arcctg(sqrt(3)/3) ⇔ 90°- arcctg(sqrt(3)/3)
- -arctg(-sqrt(3)/3)
- arcctg(1/sqrt(3)/3)
- $arcsin(\frac{\frac{\sqrt{3}}{3}}{\sqrt{(\frac{\sqrt{3}}{3})^{2}+1}})$
- $2arctg(\frac{\frac{\sqrt{3}}{3}}{1+\sqrt{(\frac{\sqrt{3}}{3})^{2}+1}})$
The infinite series of arctg sqrt(3)/3 is: $\sum_{n=0}^{\infty} \frac{(-1)^{n}(\frac{\sqrt{3}}{3})^{2n+1}}{(2n+1)}$.
Next, we discuss the derivative of arctg sqrt(3)/3 for x = sqrt(3)/3. In the following paragraph you can additionally learn what the search calculations form in the sidebar is used for.
Derivative of arctg sqrt(3)/3
The derivative of arctg sqrt(3)/3 is particularly useful to calculate the inverse tangent sqrt(3)/3 as an integral.
The formula for x is (arctg x)’ = $\frac{1}{1 + x^{2}}$, x ≠ -1,1, so for x = sqrt(3)/3 the derivative equals 0.75.
Using the arctg sqrt(3)/3 derivative, we can calculate the angle as a definite integral:
arctg sqrt(3)/3 = $\int_{0}^{\frac{\sqrt{3}}{3}}\frac{1}{{1+z^{2}}}dz$.
The relationship of arctg of sqrt(3)/3 and the trigonometric functions sin, cos and tg is:
- sin(arctangent(sqrt(3)/3)) = $\frac{\frac{\sqrt{3}}{3}}{\sqrt{1 + (\frac{\sqrt{3}}{3})^{2}}}$
- cos(arctangent(sqrt(3)/3)) = $\frac{1}{\sqrt{1 + (\frac{\sqrt{3}}{3})^{2}}}$
- tg(arctangent(sqrt(3)/3)) = sqrt(3)/3
Note that you can locate many terms including the arctangent(sqrt(3)/3) value using the search form. On mobile devices you can find it by scrolling down. Enter, for instance, arctgsqrt(3)/3 angle.
Using the aforementioned form in the same way, you can also look up terms including derivative of inverse tangent sqrt(3)/3, inverse tangent sqrt(3)/3, and derivative of arctg sqrt(3)/3, just to name a few.
In the next part of this article we discuss the trigonometric significance of arctangent sqrt(3)/3, and there we also explain the difference between the inverse and the reciprocal of tg sqrt(3)/3.
What is arctg sqrt(3)/3?
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, and 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, sin α = y / r = y, cos α = x / r = x, and tg α = y / x = sqrt(3)/3. The angle whose tangent value equals sqrt(3)/3 is α.
In the interval ]-π/2, π/2[ or ]-90°, 90°[, there is only one α whose arctangent value equals sqrt(3)/3. For that interval we define the function which determines the value of α as
From the definition of arctg(sqrt(3)/3) follows that the inverse function y-1 = tg(y) = sqrt(3)/3. Observe that the reciprocal function of tg(y),(tg(y))-1 is 1/tg(y).
Avoid misconceptions and remember (tg(y))-1 = 1/tg(y) ≠ tg-1(y) = arctg(sqrt(3)/3). And make sure to understand that the trigonometric function y=arctg(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 = arctg(x) if and only if tg y = x and -π/2 < y < π/2. The domain of x is $\mathbb{R}$.
Conclusion
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The frequently asked questions in the context include what is arctg sqrt(3)/3 degrees and what is the inverse tangent sqrt(3)/3 for example; reading our content they are no-brainers.
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– Article written by Mark, last updated on February 6th, 2017