# Trigonometry/Polar

While *SohCahToa* is a popular mnemonic, it is better to define three essential trig functions using polar coordinates. The figure shows how the sine and cosine functions can be defined for all *θ*, i.e., angles outside the first quadrant (*0<θ<π/2*).

This calculates *x* and *y* if *r* and *θ* are known.

**Example:**Suppose*θ*is in the second quadrant, e.g.,*θ=3π/4*, we see that*x<0*, and hence*cos(3π/4)<0*. On the other hand,*sin(3π/4)>0*in the second quadrant, since*y>0*for this value of*θ*.

The following calculates *r* and *θ* if *x* and *y* are known:

The tangent and its inverse function are defined by

This simple relation between the tangent and its inverse holds only in the first quadrant (*0<θ<π/2*) because the inverse function is not well defined for all angles. Also,it must be emphasized that the exponent on the function does NOT represent the multiplicative inverse:

For this reason, some authors avoid the superscript "-1" and instead write the arctangent as *arctan*:

Just as lines of constant *x* and *y* are used to illustrate a Cartesian coordinate system contours of constant *r* and *θ* are used to depict polar coordinates as shown above and to the right.

In *physics* these equations are often used to describe the components of a vector. If *A _{x}* and

*A*are the components of the vector

_{y}*:*

**A**