How do you solve 27.82 equals X squared divided by Sin squared x?

Answer 1

Here is a way to solve 27.82 = X2/SIN2(X) using successive approximations or bracketing:

First, take the square root of each side: 5.27 = X/SIN(X)

SIN(X) has values from 0 to 1 in the first quadrant (0 to 90 deg.) and from 1 to 0 in the second quadrant (90 to 180 deg.) as seen in a table of trig functions.

To bracket the answer, plug in values for X in radians; 90 deg. = Pi/2 radians where Pi =3.1415. Pi/2 radians = 1.571

For values of X from 0 to Pi/2, the value of X is too small, so X > 1.571.

Since SIN(X) is negative for angles of 180 to 360 deg., the answer should lie between

X = 1.571 and X = 3.1415 (Pi radians or 180 deg.)

Next, try an answer that lies halfway between 1.571 and 3.1415:

Let X = (1.571+3.1415)/2 = 2.356 and solve for X/SIN(X) = 3.33

Since this answer is too small, X must lie between 2.356 and 3.1415

If we continue to bracket the value of X, after a few more steps, we find that

if X = 2.621, X/SIN(X) = 5.269 which rounds up to 5.27

This problem can be more easily solved by setting it up in an Excel spread sheet and simply plugging in values for X to converge on the answer.