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:
In physics these equations are often used to describe the components of a vector. If Ax and Ay are the components of the vector A: