Solution 4.3:6c

From Förberedande kurs i matematik 1

Because the angle v satisfies v23, v belongs to the third quadrant in the unit circle. Furthermore, tan v=3 gives that the line which corresponds to the angle v

v has a gradient of 3.

slope 3

In the third quadrant, we can introduce a right-angled triangle in which the hypotenuse is 1 and the sides have a 3:1 ratio.

If we now use Pythagoras' theorem on the triangle, we see that the horizontal side a satisfies

a2+3a2=12 

which gives us that

10a2=1 i.e. a=110

Thus, the angle v's x -coordinate is −110 and y -coordinate is −310, i.e.

cosv=−−110

sinv=−310