# Square Root 3/2 and 0.8660: Explained

• lorik
In summary, arc tangent of -1 is -pi/4, and you should know the following angles: 0, pi/6, pi/4, pi/3, and pi/2, 2pi/3, 3pi/4, 5pi/6, and pi. Memorizing these angles will help you with other trig functions.f

#### lorik

Question is pretty simple :
How do I know that square root 3/2 = 0.8660
or how can 0.8660 be converted into square root 3/2 more importantly. My calculator is out of style so it displays only numbers. Thanks

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I am not sure what your question is. Could you try to ask again?

Note that .8660 is NOT equal to $\frac {\sqrt 3} 2$ but is a truncated form of the full decimal number.

@integral
Yes sorry for lack of info ,is there any to know that 0.866 is actually square root 3/2 and how do i come to this conclusion ? and vice versa

Arithmetic?

more related to trigonometry
ohh nevermind I am having some difficulties with complex numbers for example arc tangent of minus square root of 3/3 =- pi/6 how can it be -pi/6 I know its inverse but my calculator does not show pi's or dividers which I will need ! Thats why I need to know the appropriate pi !

more related to trigonometry
ohh nevermind I am having some difficulties with complex numbers for example arc tangent of minus square root of 3/3 =- pi/6 how can it be -pi/6 I know its inverse but my calculator does not show pi's or dividers which I will need ! Thats why I need to know the appropriate pi !
There are a few angles that you should just know - without having to resort to a calculator. These angles are 0, pi/6, pi/4, pi/3, and pi/2, 2pi/3, 3pi/4, 5pi/6, and pi. You should memorize the sine and cosine of each of these angles, and from these you can get all the other trig functions.

The arctangent of -1 is -pi/4.

First, as integral told you, and you apparently did not understand because you immediately asked the same question again, is that there is NO way to "know that 0.866 is actually square root 3/2" because it is not true! .866 is approximately square root of 3, divided by 2. And the only way to know that is to actually take the square root of 3 to four decimal places, divide by 2, and round to three decimal places.

As for the angles Mark44 mentions- If one angle of a right triangle is 45 degrees ($\pi/4$ radians), then the other angle must be 90- 45= 45 degrees also. That means that the right triangle is "isosceles"- if two angles are the same, then the two legs are the same length. Taking that length to be 1, by the Pythagorean theorem, the hypotenuse has length $\sqrt{2}$ and it is easy to see that the $sin(45)= 1/\sqrt{2}= \sqrt{2}/2$.

An equilateral triangle, with all sides the same length, say, 1, must have all angles the same length: 180/2= 60 degrees or $\pi/3$ radians. If you drop a perpendicular from one vertex to the opposite side, it is easy to show that both the opposite side and the angle are bisecte so you have two right triangles with angles 60 degrees and 30 degrees ($\pi/6$ radians). The hypotenuse has length 1 and the side opposite the 30 degree angle has length 1/2. You can then use the Pythagorean theorem to show that the other leg, opposite the 60 degree angle, has length $\sqrt{3}/2$.

That is enough to tell you that sin(30)= 1/2, cos(30)= $\sqrt{3}/2$, sin(60)= $\sqrt{3}/2$, and cos(60)= 1/2.

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@Mark44,@HallsofIvy
Thanks for the replies ,it is obviously true what you all are saying. I hope I didnt bother much 