11.1 Graphs, trigonometric identities, and solving trigonometric  (Page 7/12)

Algebraic solution

The inverse trigonometric functions $arcsin$ , $arccos$ and $arctan$ can also be used to solve trigonometric equations. These may be shown as second functions on your calculator: sin ${}^{-1}$ , cos ${}^{-1}$ and tan ${}^{-1}$ .

Using inverse trigonometric functions, the equation

is solved as

Trigonometric equations often look very simple. Consider solving the equation $sin\theta =0,7$ . We can take the inverse sine of both sides to find that $\theta ={sin}^{-1}(0,7)$ . If we put this into a calculator we find that ${sin}^{-1}(0,7)=44,{42}^{\circ }$ . This is true, however, it does not tell the whole story.

As you can see from [link] , there are four possible angles with a sine of $0,7$ between $-{360}^{\circ }$ and ${360}^{\circ }$ . If we were to extend the range of the sine graph to infinity we would in fact see that there are an infinite number of solutions to this equation! This difficulty (which is caused by the periodicity of the sine function) makes solving trigonometric equations much harder than they may seem to be.

Any problem on trigonometric equations will require two pieces of information to solve. The first is the equation itself and the second is the range in which your answers must lie. The hard part is making sure you find all of the possible answers within the range. Your calculator will always give you the smallest answer ( i.e.  the one that lies between $-{90}^{\circ }$ and ${90}^{\circ }$ for tangent and sine and one between $0^{\circ }$ and ${180}^{\circ }$ for cosine). Bearing this in mind we can already solve trigonometric equations within these ranges.

We can, of course, solve trigonometric equations in any range by drawing the graph.