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When we solve equations , we got some intervals and sets of the answers .Easily we can make union of intervals (Put them together and create final answer ) . For example consider $x \in [-1 \ 1]$ and $x \in [-2 \ 2]$ and the final answer is $x \in [-2 \ 2]$.

Problem appears in trigonometry equations . For example if $x=2k\pi$ and $2k\pi \pm 2\pi/3$ .Therefore the final answer is $2k\pi/3$ . I know we can put numbers in these formulas and get the same result but I want a way for doing this and how we can put solutions together in similar situations (trigonometry equations ) and get the answer.

2 Answers 2

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There isn't always a nice simplification; sometimes, the best way to write the result really is in union form.

In fact, even in the example you gave, some might aesthetically prefer the solution to be written in the form $$ x = 2 \pi n \quad \text{or} \quad x = \frac{2}{3} \pi + 2 \pi n \quad \text{or} \quad x = \frac{4}{3} \pi + 2 \pi n $$ instead of the form $x = 2 \pi n/3$.

The most basic thing to do to find some simplifications is to just look at the results a little bit to see if there is anything obvious. Once you write out the first few solutions, it may be clear that the two patterns can be combined into one.

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    So , there is no mathematically method that can help us . Is this true ? – 2017-02-20
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    @S.H.W: It may be ad-hoc, but that doesn't mean it's not mathematical. There's a reason we're taught to discern patterns on sight; it's a practical and efficient way to identify things. – 2017-02-20
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    @S.H.W: There are ways to identify special forms if you're looking for them; e.g. if you hypothesize a solution set might be an arithmetic sequence, you simply check two consecutive terms and figure out what the sequence would have to be. Values of a polynomial? Make a table of differences or try polynomial interpolation. But, by hand, it's impractical to make a checklist of every form you might want to consider and how to test for it; instead, one learns to guess what forms a solution might have or to recognize patterns upon looking. – 2017-02-20
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    First way that you said is very good but I can't understand second way (value of a polynomial) – 2017-02-20
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Formulas like $2k\pi \pm 2\pi/3$ represent a discrete set of points, so you can't really use interval notation.

If you want to be really formal about it, you can use set-builder notation. Using this method, $2k\pi \pm 2\pi/3$ could be expressed as:

$$ \left\{ 2k\pi \pm \frac{2\pi}3 : k \in \Bbb Z\right\} $$

Simple example with unions: Solve $\sin x = \dfrac12$.

We know that $\sin x = \frac12$ means $x = \dfrac\pi6 + 2k\pi$ and $x = \dfrac{5\pi}6 + 2k\pi$. These two expressions can't be combined into one, so our answer would have to be a union of two sets if we wanted to be very formal about it: $$ x \in \left\{ \frac\pi6 + 2k\pi : k \in \Bbb Z\right\} \cup \left\{ \frac{5\pi}6 + 2k\pi : k \in \Bbb Z\right\} $$

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    Okay , but you didn't say how we can put together answers in trigonometry equations. – 2017-02-20
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    If you want to put together multiple answers you'll need to use unions, unless there's a way the answers can be combined into one expression involving $k$. – 2017-02-20
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    @S.H.W, see my edit/addendum. – 2017-02-20
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    Thank you , your answer is good but my problem isn't solved . In many questions the answer is only in one form . – 2017-02-20
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    @S.H.W, if it's in one form then you'll still need set-builder notation but without the union. For example, $\sin x = 0$ means $x = k\pi$. Then formally the answer would be $x \in \{ k\pi : k \in \Bbb Z\}$. There's no way to do it with intervals. – 2017-02-20