Table of Contents
Welcome to arccsc sqrt2 – sqrt6, our post aboutthe arccosecant of sqrt2 – sqrt6.
For the inverse trigonometric function of arccsc sqrt2 – sqrt6 we usually employ the abbreviation arccsc and write it as arccsc sqrt2 – sqrt6 or arccsc(sqrt2 – sqrt6).
If you have been looking for what is arccsc sqrt2 – sqrt6, either in degrees or radians, or if you have been wondering about the inverse of csc sqrt2 – sqrt6, then you are right here, too.
In this post you can find the angle arccosecant of sqrt2 – sqrt6, along with identities.
Read on to learn all about the arccsc of sqrt2 – sqrt6, and note that the term sqrt2-sqrt6 is approximately -1.03527618 as a decimal number.
Arccsc of sqrt2 – sqrt6
If you want to know what is arccsc sqrt2 – sqrt6 in terms of trigonometry, check out the explanations in the last paragraph; ahead in this section is the value of arccosecant(sqrt2 – sqrt6):
arccosecant sqrt2 – sqrt6 = -5pi/12 rad = -75 °
arccosecant of sqrt2 – sqrt6 = -5pi/12 radians = -75 degrees
The arccsc of sqrt2 – sqrt6 is -5pi/12 radians, and the value in degrees is -75°. To change the result from the unit radian to the unit degree multiply the angle by 180° / $\pi$ and obtain -75°.
Our results above contain fractions of pi for the results in radian, and are exact values otherwise. If you compute arccsc(sqrt2 – sqrt6), 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 sqrt2 – sqrt6 as decimal in the field labelled “x”. However, if you want to be given the angle of csc sqrt2 – sqrt6 in radians, then you must press the swap units button.
Calculate arccsc x
The identities of arccosecant sqrt2 – sqrt6 are as follows: arccsc(sqrt2 – sqrt6) =
- $\frac{\pi}{2}$ – arcsec(sqrt2 – sqrt6) ⇔ 90°- arcsec(sqrt2 – sqrt6)
- -arccsc(-sqrt2 – sqrt6)
- arcsin(1/sqrt2 – sqrt6)
The infinite series of arccsc sqrt2 – sqrt6 is: $\sum_{n=0}^{\infty}\frac{\binom{2n}{n}(\sqrt{2}-\sqrt{6})^{-(2n+1)}}{4^{n}(2n+1)}$.
Next, we discuss the derivative of arccsc sqrt2 – sqrt6 for sqrt2 – sqrt6 = sqrt2 – sqrt6. In the following paragraph you can additionally learn what the search calculations form in the sidebar is used for.
Derivative of arccsc sqrt2 – sqrt6
The derivative of arccsc sqrt2 – sqrt6 is particularly useful to calculate the inverse arccsc sqrt2 – sqrt6 as an integral.
The formula for x is (arccsc x)’ = -$\frac{1}{|x|\sqrt{x^{2}-1}}$, |x| > 1, so for x = sqrt2 – sqrt6 the derivative equals -3.6048842601.
Using the arccsc sqrt2 – sqrt6 derivative, we can calculate the angle as a definite integral:
arccsc sqrt2 – sqrt6 = $\int_{\sqrt{2}-\sqrt{6}}^{\infty}\frac{1}{z\sqrt{z^{2}-1}}dz$.
The relationship of arccsc of sqrt2 – sqrt6 and the trigonometric functions sin, cos and tan is:
- sin(arccosecant(sqrt2 – sqrt6)) = $\frac{1}{(\sqrt{2}-\sqrt{6})}$
- cos(arccosecant(sqrt2 – sqrt6)) = $\frac{\sqrt{(\sqrt{2}-\sqrt{6})^{2}-1}}{\sqrt{2}-\sqrt{6}}$
- tan(arccosecant(sqrt2 – sqrt6)) = $\frac{1}{\sqrt{(\sqrt{2}-\sqrt{6})^{2}-1}}$
Note that you can locate many terms including the arccosecant(sqrt2 – sqrt6) value using the search form. On mobile devices you can find it by scrolling down. Enter, for instance, arccscsqrt2 – sqrt6 angle.
Using the aforementioned form in the same way, you can also look up terms including derivative of inverse arccsc sqrt2 – sqrt6, inverse arccsc sqrt2 – sqrt6, and derivative of arccsc sqrt2 – sqrt6, just to name a few.
In the next part of this article we discuss the trigonometric significance of arccosecant sqrt2 – sqrt6, and there we also explain the difference between the inverse and the reciprocal of csc sqrt2 – sqrt6.
What is arccsc sqrt2 – sqrt6?
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, sin α = y / r = y, and csc α = 1 / y = sqrt2 – sqrt6. The angle whose arccsc value equals sqrt2 – sqrt6 is α.
In the interval [-pi/2, 0[ ∪ ]0, pi/2] or [-90°, 0[ ∪ ]0, 90°], there is only one α whose sine value equals sqrt2 – sqrt6. For that interval we define the function which determines the value of α as
From the definition of arccsc(sqrt2 – sqrt6) follows that the inverse function y-1 = csc(y) = sqrt2 – sqrt6. Observe that the reciprocal function of csc(y),(csc(y))-1 is 1/csc(y) = sin(y).
Avoid misconceptions and remember (csc(y))-1 = 1/csc(y) ≠ csc-1(y) = arccsc(sqrt2 – sqrt6). And make sure to understand that the trigonometric function y=arccsc(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 = arccsc(x) if and only if csc y = x and -pi/2 ≤ y < 0 or csc y = x and 0 < y ≤ pi/2. The domain of x is x ≤ −1 or 1 ≤ x.
Conclusion
The frequently asked questions in the context include what is arccsc sqrt2 – sqrt6 degrees and what is the inverse arccsc sqrt2 – sqrt6 for example; reading our content they are no-brainers.
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– Article written by Mark, last updated on February 4th, 2017