Lösung 4.4:8c

Aus Online Mathematik Brückenkurs 1

When we have a trigonometric equation which contains a mixture of different trigonometric functions, a useful strategy can be to rewrite the equation so that it is expressed in terms of just one of the functions. Sometimes, it is not easy to find a way to rewrite it, but in the present case a plausible way is to replace the “1” in the numerator of the left-hand side with sin2x+cos2x using the Pythagorean identity. This means that the equation's left-hand side can be written as

and the expression is then completely expressed in terms of tan x,

If we substitute t=tanx, we see that we have a quadratic equation in t, which, after simplifying, becomes t2+t=0 and has roots t=0 and t=−1. There are therefore two possible values for tanx, tanx=0 or tanx=−1. The first equality is satisfied when x=n for all integers n, and the second when x=34+n.

The complete solution of the equation is

xx=n=43+n

where n is an arbitrary integer.