Table of Contents
Welcome to arcsec -sqrt2, our post aboutthe arcsecant of -sqrt2.
For the inverse trigonometric function of secant -sqrt2 we usually employ the abbreviation arcsec and write it as arcsec -sqrt2 or arcsec(-sqrt2).
If you have been looking for what is arcsec -sqrt2, either in degrees or radians, or if you have been wondering about the inverse of sec -sqrt2, then you are right here, too.
In this post you can find the angle arcsecant of -sqrt2, along with identities.
Read on to learn all about the arcsec of -sqrt2, and note that the term -sqrt2 is approximately -1.41421356 as a decimal number.
Arcsec of -sqrt2
If you want to know what is arcsec -sqrt2 in terms of trigonometry, check out the explanations in the last paragraph; ahead in this section is the value of arcsecant(-sqrt2):
arcsecant -sqrt2 = 3pi/4 rad = 135 °
arcsecant of -sqrt2 = 3pi/4 radians = 135 degrees
The arcsec of -sqrt2 is 3pi/4 radians, and the value in degrees is 135°. To change the result from the unit radian to the unit degree multiply the angle by 180° / $\pi$ and obtain 135°.
Our results above contain fractions of pi for the results in radian, and are exact values otherwise. If you compute arcsec(-sqrt2), 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 as decimal in the field labelled “x”. However, if you want to be given the angle of sec -sqrt2 in radians, then you must press the swap units button.
Calculate arcsec x
The identities of arcsecant -sqrt2 are as follows: arcsec(-sqrt2) =
- $\frac{\pi}{2}$ – arccsc(-sqrt2) ⇔ 90°- arccsc(-sqrt2)
- $\pi$ – arcsec(sqrt2) ⇔ 180° – arcsec(sqrt2)
- arccos($\frac{1}{-\sqrt{2}}$)
The infinite series of arcsec -sqrt2 is: $\frac{\pi}{2} – \sum_{n=0}^{\infty}\frac{\binom{2n}{n}(-\sqrt{2})^{-(2n+1)}}{4^{n}(2n+1)}$.
Next, we discuss the derivative of arcsec -sqrt2 for -sqrt2 = -sqrt2. In the following paragraph you can additionally learn what the search calculations form in the sidebar is used for.
Derivative of arcsec -sqrt2
The derivative of arcsec -sqrt2 is particularly useful to calculate the inverse secant -sqrt2 as an integral.
The formula for x is (arcsec x)’ = $\frac{1}{|x|\sqrt{x^{2}-1}}$, |x| > 1, so for x = -sqrt2 the derivative equals 0.7071067812.
Using the arcsec -sqrt2 derivative, we can calculate the angle as a definite integral:
arcsec -sqrt2 = $\int_{1}^{-\sqrt{2}}\frac{1}{z\sqrt{z^{2}-1}}dz$.
The relationship of arcsec of -sqrt2 and the trigonometric functions sin, cos and tan is:
- sin(arcsecant(-sqrt2)) = $\frac{\sqrt{(-\sqrt{2})^{2}-1}}{-\sqrt{2}}$
- cos(arcsecant(-sqrt2)) = $\frac{1}{-\sqrt{2}}$
- tan(arcsecant(-sqrt2)) = $\sqrt{(-\sqrt{2})^{2}-1}$
Note that you can locate many terms including the arcsecant(-sqrt2) value using the search form. On mobile devices you can find it by scrolling down. Enter, for instance, arcsec-sqrt2 angle.
Using the aforementioned form in the same way, you can also look up terms including derivative of inverse secant -sqrt2, inverse secant -sqrt2, and derivative of arcsec -sqrt2, just to name a few.
In the next part of this article we discuss the trigonometric significance of arcsecant -sqrt2, and there we also explain the difference between the inverse and the reciprocal of sec -sqrt2.
What is arcsec -sqrt2?
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, cos α = x / r = x, and sec α = 1 / x = -sqrt2. The angle whose secant value equals -sqrt2 is α.
In the interval [0, pi/2[ ∪ ]pi/2, pi] or [0°, 90°[ ∪ ]90°, 180°], there is only one α whose secant value equals -sqrt2. For that interval we define the function which determines the value of α as
From the definition of arcsec(-sqrt2) follows that the inverse function y-1 = sec(y) = -sqrt2. Observe that the reciprocal function of sec(y),(sec(y))-1 is 1/sec(y) = cos(y).
Avoid misconceptions and remember (sec(y))-1 = 1/sec(y) ≠ sec-1(y) = arcsec(-sqrt2). And make sure to understand that the trigonometric function y=arcsec(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 = arcsec(x) if and only if sec y = x and 0 ≤ y < pi/2 or sec y = x and pi/2 < y ≤ pi. The domain of x is x ≤ −1 or 1 ≤ x.
Conclusion
The frequently asked questions in the context include what is arcsec -sqrt2 degrees and what is the inverse secant -sqrt2 for example; reading our content they are no-brainers.
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– Article written by Mark, last updated on February 4th, 2017