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Solution 4.4:4

From Förberedande kurs i matematik 1

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The idea is first to find the general solution to the equation and then to see which angles lie between 0 and 360.

If we start by considering the expression 2v+10 as an unknown, then we have a usual basic trigonometric equation. One solution which we can see directly is


2v+10=110


There is then a further solution which satisfies 02v+10360, where 2v+10 lies in the third quadrant and makes the same angle with the negative y-axis as 100 makes with the positive y -axis, i.e. 2v+10 makes an angle 11090=20  with the negative y -axis and consequently


2v+10=27020=250



There is then a further solution which satisfies 02v+10360, where 2v+10 lies in the third quadrant and makes the same angle with the negative y-axis as 100 makes with the positive y -axis, i.e. 2v+10 makes an angle 11090=20  with the negative y -axis and consequently


2v+10=27020=250


FIGURE1 FIGURE2

Now it is easy to write down the general solution,


2v+10=110+n360 and

2v+10=250+n360


and if we make v the subject, we get


v=50+n180 and

v=120+n180 EQ6

For different values of the integers n, we see that the corresponding solutions are:


n=2n=1n=0n=1n=2n=3v=502180=310v=501180=130v=50+0180=50v=50+1180=230v=50+2180=410v=50+3180=590v=1202180=240v=1201180=60v=120+0180=120v=120+1180=300v=120+2180=480v=120+3180=660


From the table, we see that the solutions that are between 0 and 360 are


v=50v=120v=230 and v=300