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Hello I am not sure about one thing about trigonometrics equations. Exercise is to solve the equation: $\cos(x) = 1/2$

Formula to count cos(x)=a
x = x0  + 2kπ or x = -x0  + 2kπ 
where cos(x0) = a ,k is integer number

My Answers:
x0= π/3
x=π/3 + 2kπ or x = -π/3 +2kπ ,
k is an integer number

Calculator's Answers
x=π/3 + 2kπ or x = 5π/3 +2kπ ,
k is an integer number

the only difference is -π/3 and 5π/3(coterminal angles)
My question is, can I write answer like this:

x=π/3 + 2kπ or x = -π/3 +2kπ

or should I change negative values to coterminal angles to look like this:

x=π/3 + 2kπ or x = 5π/3 +2kπ**
Suzu Hirose
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Paweł Michoń
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2 Answers2

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In general, I would prefer $x = 2k\pi + \frac{\pi}{3} \; \text{or} \; x = 2k\pi - \frac{\pi}{3}, \, k \in \mathbb{Z}$ for $2$ reasons:

$1)$ It is a more simplified version.
$2)$ It can be written as $x = 2k\pi \pm \frac{\pi}{3}, \, k \in \mathbb{Z}$ which is concise.

That being said, both versions are correct. You can also write

$$x = 2k\pi + \frac{11\pi}{3}, \; x = 2k\pi + \frac{17\pi}{3}, \; x = 2k\pi + \frac{23\pi}{3}, ...$$

What is important is the right balance between simplicity and oblivion.

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It really doesn't matter. The only important part is that you have found one of the primitive solutions and also given an explicit formula for every other solution. Typically, one takes the smallest positive solution of a periodic function, so I would write:

$$x = \frac{\pi}{3}+k\cdot2\pi, \quad k\in\mathbb{Z}$$

Nuke_Gunray
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