# After what interval in degrees or radians do sine, cosine and tangent values repeat?

Between $0$ to $2π$, I have noticed that $\sin x$, $\cos x$ and $\tan x$ values repeat for different values of $x$.

For example, $\sin 30 = \sin 150$

What exactly is the interval between two successive values of $x$ such that the value of $\sin x$, $\cos x$ or $\tan x$ are equal?

• There are several ways of doing this. Have you tried drawing out their graphs? Do you know the geometric meaning of $\sin, \cos, \tan$? Aug 30, 2013 at 16:19
• As indicated in the answer below, the sine and cosine repeat every $360^{\circ}$, and the tangent repeats every $180^{\circ}$. These are called the periods of these functions. Aug 30, 2013 at 17:30
• Please, please, please, use the degree symbol "^\circ" if you want degrees. If you don't use it, you mean radians, whether that's what you want or not. Aug 30, 2013 at 23:10

$\sin(x) = \sin(\pi - x)$, or, equivalently
$\sin(x) = \sin(180^o - x)$.

$\sin(x) = \sin(2\pi n + x) \forall n \in \mathbb{Z}$, or, equivalently,
$\sin(x) = \sin(360^o n + x) \forall n \in \mathbb{Z}$.

$\cos(x) = \cos(-x)$. Alternately,
$\cos(x) = \cos(360^o - x)$, or (equivalently to the latter)
$\cos(x) = \cos(2\pi - x)$.

$\cos(x) = \cos(2\pi n + x) \forall n \in \mathbb{Z}$, or, equivalently
$\cos(x) = \cos(360^o n + x) \forall n \in \mathbb{Z}$.

$\tan(x) = \tan(\pi + x)$, or, equivalently,
$\tan(x) = \tan(180^o + x)$.

$\tan(x) = \tan(\pi n + x) \forall n \in \mathbb{Z}$, or, equivalently,
$\tan(x) = \tan(180^o n + x) \forall n \in \mathbb{Z}$.

• I had to downvote this because of the formatting. It's unreadable to me in its current form. Aug 30, 2013 at 18:25
• What makes it unreadable? Aug 30, 2013 at 18:39
• It's a giant jumbled mess with too many equations crammed too close together. Moreover, half of the equations say the same thing (e.g., the very first line). Aug 30, 2013 at 18:47
• Tried to make it better. Half the equations say the same thing because OP wasn't very clear on whether to use degrees or radians, so I just gave both. I'm not really sure what you want from the formatting, but I'd hate for it to be so bad that you can't read it at all. Aug 30, 2013 at 18:53
• Much better. I think tables work best in these cases, but I know that there is only so much time one is willing to spend on these answers (and rightfully so, too). Aug 30, 2013 at 18:56

$1:$

Let $\sin x=\sin y$

Applying $\sin C-\sin D=2\sin\frac{C-D}2\cos\frac{C+D}2,$

we have $2\sin\frac{x-y}2\cos\frac{x+y}2=0$

If $\sin\frac{x-y}2=0, \frac{x-y}2=n180^\circ\implies x-y=n180^\circ$ where $n$ is any integer

If $\cos\frac{x+y}2=0, \frac{x+y}2=(2m+1)90^\circ\implies x+y=(2m+1)180^\circ$ where $m$ is any integer

$2:$

Apply $\cos C-\cos D=-2\sin\frac{C-D}2\sin\frac{C+D}2,$

$3:$

$\displaystyle\tan x=\tan y\iff \frac{\sin x}{\cos x}=\frac{\sin y}{\cos y}$

$$\implies \sin x\cos y-\cos x\sin y=0\implies \sin(x-y)=0$$

$\implies x-y=r180^\circ$ where $r$ is any integer