Table of Contents
Welcome to arcsin -sqrt(3)/2, our post aboutthe arcsine of -sqrt(3)/2.
For the inverse trigonometric function of sine -sqrt(3)/2 we usually employ the abbreviation arcsin and write it as arcsin -sqrt(3)/2 or arcsin(-sqrt(3)/2).
If you have been looking for what is arcsin -sqrt(3)/2, either in degrees or radians, or if you have been wondering about the inverse of sin -sqrt(3)/2, then you are right here, too.
In this post you can find the angle arcsine of -sqrt(3)/2, along with identities.
Read on to learn all about the arcsin of -sqrt(3)/2, and note that the term -sqrt(3)/2 is approximately -0.8660254 as a decimal number.
Arcsin of -sqrt(3)/2
If you want to know what is arcsin -sqrt(3)/2 in terms of trigonometry, check out the explanations in the last paragraph; ahead in this section is the value of arcsine(-sqrt(3)/2):
arcsine -sqrt(3)/2 = -pi/3 rad = -60 °
arcsine of -sqrt(3)/2 = -pi/3 radians = -60 degrees
The arcsin of -sqrt(3)/2 is -pi/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 pi for the results in radian, and are exact values otherwise. If you compute arcsin(-sqrt(3)/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(3)/2 as decimal in the field labelled “x”. However, if you want to be given the angle opposite to -sqrt(3)/2 in radians, then you must press the swap units button.
Calculate arcsin x
The identities of arcsine -sqrt(3)/2 are as follows: arcsin(-sqrt(3)/2) =
- $\frac{\pi}{2}$ – arcscos(-sqrt(3)/2) ⇔ 90°- arcscos(-sqrt(3)/2)
- -arcsin(sqrt(3)/2)
- arccsc(1/-sqrt(3)/2)
- $\frac{arccos(1-2(-\frac{\sqrt{3}}{2})^{2})}{2}$
The infinite series of arcsin -sqrt(3)/2 is: $\sum_{n=0}^{\infty} \frac{(2n)!}{2^{2n}(n!)^{2}(2n+1)}(-\frac{\sqrt{3}}{2})^{2n+1}$.
Next, we discuss the derivative of arcsin x for x = -sqrt(3)/2. In the following paragraph you can additionally learn what the search calculations form in the sidebar is used for.
Derivative of arcsin -sqrt(3)/2
The derivative of arcsin -sqrt(3)/2 is particularly useful to calculate the inverse sine -sqrt(3)/2 as an integral.
The formula for x is (arcsin x)’ = $\frac{1}{\sqrt{1-x^{2}}}$, x ≠ -1,1, so for x = -sqrt(3)/2 the derivative equals 2.
Using the arcsin -sqrt(3)/2 derivative, we can calculate the angle as a definite integral:
arcsin -sqrt(3)/2 = $\int_{0}^{-\frac{\sqrt{3}}{2}}\frac{1}{\sqrt{1-z^{2}}}dz$.
The relationship of arcsin of -sqrt(3)/2 and the trigonometric functions sin, cos and tan is:
- sin(arcsine(-sqrt(3)/2)) = -sqrt(3)/2
- cos(arcsine(-sqrt(3)/2)) = $\sqrt{1 – (-\frac{\sqrt{3}}{2})^{2}}$
- tan(arcsine(-sqrt(3)/2)) = $\frac{-\frac{\sqrt{3}}{2}}{\sqrt{1 – (-\frac{\sqrt{3}}{2})^{2}}}$
Note that you can locate many terms including the arcsine(-sqrt(3)/2) value using the search form. On mobile devices you can find it by scrolling down. Enter, for instance, arcsin-sqrt(3)/2 angle.
Using the aforementioned form in the same way, you can also look up terms including derivative of inverse sine -sqrt(3)/2, inverse sine -sqrt(3)/2, and derivative of arcsin -sqrt(3)/2, just to name a few.
In the next part of this article we discuss the trigonometric significance of arcsine -sqrt(3)/2, and there we also explain the difference between the inverse and the reciprocal of sin -sqrt(3)/2.
What is arcsin -sqrt(3)/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(3)/2 / r = -sqrt(3)/2. The angle whose sine value equals -sqrt(3)/2 is α.
In the interval [-pi/2, pi/2] or [-90°, 90°], there is only one α whose sine value equals -sqrt(3)/2. For that interval we define the function which determines the value of α as
From the definition of arcsin(-sqrt(3)/2) follows that the inverse function y-1 = sin(y) = -sqrt(3)/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(3)/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
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– Article written by Mark, last updated on February 4th, 2017