Inverses of Trig Functions


When solving equations involving trig functions, you'll need to use their inverse functions. This is the same as knowing that to inverse a multiplication you must divide.


The inverses of Sin, Cos and Tan can be expressed in two interchangable ways: \[\text{Inverse Sin: }\sin^{-1}(x)\hspace{6pt}\text{or }\arcsin(x)\] \[\text{Inverse Cos: }\cos^{-1}(x)\hspace{6pt}\text{or }\arccos(x)\] \[\text{Inverse Tan: }\tan^{-1}(x)\hspace{6pt}\text{or }\arctan(x)\] The first of these could be confusing if you're familiar with using exponents. In short, normally a negative power means to take the reciprocal, for example \[x^{-2}=\frac{1}{x^2}\] But this is not the case with trig functions. \[\sin^{-1}(x)\neq\frac{1}{\sin(x)}\] Click here to find out about reciprocals of trig functions.

Principal values

Because of the trig functions' periodic nature, the inverse of any given value doesn't only have one solution. For instance \[\sin^{-1}(0)=0+180n\hspace{6pt}\text{Where n is any integer}\] \[\sin^{-1}(0)=\{...,-360,-180,0,180,360,...\}\] This is why we use the principal values. These are the values from the ranges stated below, such that there is only one solution for every inverse. These are the ranges your calculator would give you an answer within if you used an inverse trig function.

\[-90\leq\sin^{-1}(x)\leq 90\]
\[0\leq\cos^{-1}(x)\leq 180\]
\[-90\leq\tan^{-1}(x)\leq 90\]

For more information about inverses and principal values,
look on this page about how to solve trig problems.