Solution 4.4:5b

From Förberedande kurs i matematik 1

Let's first investigate when the equality

is satisfied. Because tanu can be interpreted as the slope (gradient) of the line which makes an angle u with the positive x-axis, we see that for a fixed value of tanu, there are two angles v in the unit circle with this slope,

v=uandv=u+.

The angle v has the same slope after every half turn, so if we add multiples of to u, we will obtain all the angles v which satisfy the equality

v=u+n

where n is an arbitrary integer.

If we apply this result to the equation

we see that the solutions are given by

4x=x+n(n is an arbitrary integer),

and solving for x gives

x=31n(n is an arbitrary integer).