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 &theta;, i.e., angles outside the first quadrant (0<&theta;<&pi;/2).

This calculates x and y if r and &theta; are known.
 * Example: Suppose &theta; is in the second quadrant, e.g., &theta;=3&pi;/4, we see that x<0, and hence cos(3&pi;/4)<0. On the other hand, sin(3&pi;/4)>0 in the second quadrant, since y>0 for this value of &theta;.

The following calculates r and &theta; 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<&theta;<&pi;/2) because the inverse function is not well defined for all angles. Also,it must be emphasized that the exponent $$-1$$ on the $$\tan$$ function does NOT represent the multiplicative inverse:


 * $$\tan^{-1}\theta\neq\frac{1}{\tan\theta}$$

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


 * $$\theta = \tan^{-1}\frac y x \equiv \arctan \frac y x$$

Just as lines of constant x and y are used to illustrate a Cartesian coordinate system contours of constant r and &theta; 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 Ax and Ay are the components of the vector A: