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From Förberedande kurs i matematik 1

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 110−90=20 with the negative y-axis and consequently

2v+10=270−20=250.

Now it is easy to write down the general solution,

2v+102v+10=110+n360and=250+n360

and if we make v the subject, we get

vv=50+n180and=120+n180

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

n=−2:		v=50−2180=−310		v=120−2180=−240
n=−1:		v=50−1180=−130		v=120−1180=−60
n=0:		v=50+0180=50		v=120+0180=120
n=1:		v=50+1180=230		v=120+1180=300
n=2:		v=50+2180=410		v=120+2180=480
n=3:		v=50+3180=590		\displaystyle v = 120^{\circ} + 3\cdot 180^{\circ} = 660^{\circ}
\displaystyle \cdots\cdots		\displaystyle \cdots\cdots		\displaystyle \cdots\cdots

From the table, we see that the solutions that are between \displaystyle 0^{\circ} and \displaystyle 360^{\circ} are

\displaystyle v = 50^{\circ},\quad v=120^{\circ },\quad v=230^{\circ}\quad\text{and}\quad v=300^{\circ}\,\textrm{.}