trigonometry equation

Question

Solve for x in terms of y: y=cot2x

Looks like a simple equation, but I've been racking my brain trying to figure it out.

Thanks!

Answer 1

The key to this is to make a right triangle. Remember SOH CAH TOA?

If I could draw here, this would be much easier to show...

Draw a right triangle with the right angle in the lower left corner, and the bottom edge horizontal.

y=cot(2x) is the same as

y

--- =cot(2x)

1

Because the tangent is opposite over adjacent, the COTANGENT is Adjacent over Opposite.

So circle the vertex on the bottom right. This angle is your (2x). The adjacent side (the horizontal edge between your circled angle and the right angle) has a value of "y", and the vertical edge is the opposite side, it gets the value of "1". This makes the hypotenuse have a length of sqrt(1 + y²).

Now you can use the cos(2x) or the sin(2x) or even the tan(2x), because we have inverse functions for these.

If I use cos(2x) = y/sqrt(1+y²), so 2x =cos^-1(y/sqrt(1+y²)),

or x=(1/2)cos^-1(y/sqrt(1+y²))

If I use sin(2x) = 1/sqrt(1+y²), so 2x = sin^-1(1/sqrt(1+y²))

or x=(1/2)sin^-1(1/sqrt(1+y²))

If I use tan(2x) = 1/y, then 2x=tan^-1(1/y)

or x=(1/2)tan^-1(1/y)

Answer 2

y = cot(2x)

cot^-1(y) = 2x

(1/2)cot^-1(y) = x