given a tangent value, [ tan(x) ], how do you solve for x without a calculator? eg: tan[1/sqrt(3)] = ?

Answer 1

Without a calculator, you will need to know your trig definitions and identities well and also know the values of sin and cos at special angles (or know how to use the Pythagorean Theorem to determine those values).

So, you need to know first of all that tanx=sinx/cosx

You will also need to know that:
sin0=0
sin30=1/2
sin45=sqrt[2]/2
sin60=sqrt[3]/2
sin 90 =1

and
cos0=1
cos30=sqrt[3]/2
cos 45=sqrt[2]/2
cos60=1/2
cos90=0

so you can recognize that 1/sqrt[3] corresponds to tan30, since tan30=sin30/cos30
=1/2/sqrt[3]/2 = 1/sqrt[3]

we know that tanx=1 corresponds to x=45
and tan 60=sqrt[3]

hope this gets you started

Answer 2

If you construct an equilateral triangle with sides = 2, and drop a perpendicular down from the top, you bisect the base, and create two triangles. For either of the triangles, the hypotenuse is 2, the base is 1 (because it is bisected by the line you drew), and therefore the line you drew (by Pythagoras) is √3. The tan of the bisected angle (30 degrees) is 1/√3.

This is how I remember all my trig identities for 30 degrees and 60 degrees.

You can also do a triangle for 45 degrees.

Answer 3

you draw a triangle that will have sides 2, 2 ,2 and is equilateral. Then draw a line down the middle and find the length of each side by pythagoruses thereom. The apply the fact that tan = opp over adj and see which angles tan is 1/root 3

Answer 4

There is no way in general to hand calculate x from tan(x). But there are certain angles whose trig functions you should memorize, because they come up so often in mathematics. These are

0, 30, 45, 60, 90 degrees.

In radians, these angles are

0, pi/6, pi/4, pi/3, pi/2

From these you can easily get
120, 135, 150, 180
210, 225, 240, 270
300, 315, 330, 360
just be changing some positive numbers to negatives.

Not too many numbers are worth memorizing in math, but these are.

Answer 5

For trig, you have X, Y and H, where H is the hypotenuse
Since tangent is Y/X you know that X = sqrt(3) and Y is 1.
Pythagoras tells us that X^2 + y^2 = H^2
and we can figure out that H is 2.

Since sine = Y/H , we know that sin(X) = .5

Everyone is expected to know the X for which sin(x) = .5

Answer 6

2^(x-3) = 17/5 x-3 = log(3.4)/log(2) x = log(3.4)/log(2) + 3 the single way that i understand of to clean up this without employing a calculator is to scrupulously graph the two edge of the time-honored equation. in distinctive words, graph y = 5*2^(x-3) and y = 17. One has to elect uncomplicated values for x and to be very meticulous in graphing. The intersection of the two graphs may be the answer, even even with the undeniable fact which you will now no longer be geared up to get a very precise answer, through fact the answer seems to be an irrational volume.