三角函数/常用角度参考

注意：表格中某些值以包含分母中根号的形式给出 - 这样做是为了简化对倒数对的识别，以及因为表格中给出的形式在某种意义上更简单。还请注意，所有列出的特殊点的三角函数的绝对值都包含在第一象限（从 0 到 90° 或 ${\frac {\pi }{2}}$ 弧度，含边界）；所有其他值都是通过水平或垂直轴的对称性推导出来的，或者通过交换轴（在三角圆上）使得一个三角函数也与其余函数交换。

$\theta$ (正)		$\sin(\theta )$	$\cos(\theta )$	$\tan(\theta )$	$\cot(\theta )$	$\sec(\theta )$	$\csc(\theta )$	$\theta$ (负)
(度)	(弧度)							(度)	(弧度)
$0^{\circ }$	$0$	$0$	$1$	$0$	未 定义	$1$	未 定义	$-360^{\circ }$	$-2\pi$
$15^{\circ }$	${\frac {\pi }{12}}$	${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	$2-{\sqrt {3}}$	$2+{\sqrt {3}}$	${\sqrt {6}}-{\sqrt {2}}$	${\sqrt {6}}+{\sqrt {2}}$	$-345^{\circ }$	$-{\frac {13\pi }{12}}$
$22.5^{\circ }$	${\frac {\pi }{8}}$	${\frac {\sqrt {2-{\sqrt {2}}}}{2}}$	${\frac {\sqrt {2+{\sqrt {2}}}}{2}}$	${\sqrt {2}}-1$	${\sqrt {2}}+1$	${\sqrt {4-2{\sqrt {2}}}}$	${\sqrt {4+2{\sqrt {2}}}}$	$-337.5^{\circ }$	$-{\frac {15\pi }{8}}$
$30^{\circ }$	${\frac {\pi }{6}}$	${\frac {1}{2}}$	${\frac {\sqrt {3}}{2}}$	${\frac {1}{\sqrt {3}}}$	${\sqrt {3}}$	${\frac {2}{\sqrt {3}}}$	$2$	$-330^{\circ }$	$-{\frac {11\pi }{6}}$
$45^{\circ }$${\frac {\pi }{4}}$	${\frac {1}{\sqrt {2}}}$		$1$		${\sqrt {2}}$		$-315^{\circ }$	$-{\frac {7\pi }{4}}$
$60^{\circ }$	${\frac {\pi }{3}}$	${\frac {\sqrt {3}}{2}}$	${\frac {1}{2}}$	${\sqrt {3}}$	${\frac {1}{\sqrt {3}}}$	$2$	${\frac {2}{\sqrt {3}}}$	$-300^{\circ }$	$-{\frac {5\pi }{3}}$
$67.5^{\circ }$	${\frac {3\pi }{8}}$	${\frac {\sqrt {2+{\sqrt {2}}}}{2}}$	${\frac {\sqrt {2-{\sqrt {2}}}}{2}}$	${\sqrt {2}}+1$	${\sqrt {2}}-1$	${\sqrt {4+2{\sqrt {2}}}}$	${\sqrt {4-2{\sqrt {2}}}}$	$-292.5^{\circ }$	$-{\frac {11\pi }{8}}$
$75^{\circ }$	${\frac {5\pi }{12}}$	${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	$2+{\sqrt {3}}$	$2-{\sqrt {3}}$	${\sqrt {6}}+{\sqrt {2}}$	${\sqrt {6}}-{\sqrt {2}}$	$-285^{\circ }$	$-{\frac {19\pi }{12}}$
$90^{\circ }$		$1$	$0$	未 定义	$0$	未 定义	$1$	$-270^{\circ }$	$-{\frac {3\pi }{2}}$
$105^{\circ }$	${\frac {7\pi }{12}}$	${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	$-{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	$-2-{\sqrt {3}}$	$-2+{\sqrt {3}}$	$-{\sqrt {6}}-{\sqrt {2}}$	${\sqrt {6}}-{\sqrt {2}}$	$-255^{\circ }$	$-{\frac {17\pi }{12}}$
$112.5^{\circ }$	${\frac {5\pi }{8}}$	${\frac {\sqrt {2+{\sqrt {2}}}}{2}}$	$-{\frac {\sqrt {2-{\sqrt {2}}}}{2}}$	$-{\sqrt {2}}-1$	$-{\sqrt {2}}+1$	$-{\sqrt {4+2{\sqrt {2}}}}$	${\sqrt {4-2{\sqrt {2}}}}$	$-247.5^{\circ }$	$-{\frac {11\pi }{8}}$
$120^{\circ }$	${\frac {2\pi }{3}}$	${\frac {\sqrt {3}}{2}}$	$-{\frac {1}{2}}$	$-{\sqrt {3}}$	$-{\frac {1}{\sqrt {3}}}$	$-2$	${\frac {2}{\sqrt {3}}}$	$-240^{\circ }$	$-{\frac {4\pi }{3}}$
$135^{\circ }$	${\frac {3\pi }{4}}$	${\frac {1}{\sqrt {2}}}$	$-{\frac {1}{\sqrt {2}}}$	$-1$		$-{\sqrt {2}}$	${\sqrt {2}}$	$-225^{\circ }$	$-{\frac {5\pi }{4}}$
$150^{\circ }$	${\frac {5\pi }{6}}$	${\frac {1}{2}}$	$-{\frac {\sqrt {3}}{2}}$	$-{\frac {1}{\sqrt {3}}}$	$-{\sqrt {3}}$	$-{\frac {2}{\sqrt {3}}}$	$2$	$-210^{\circ }$	$-{\frac {7\pi }{6}}$
$157.5^{\circ }$	${\frac {7\pi }{8}}$	${\frac {\sqrt {2-{\sqrt {2}}}}{2}}$	$-\frac{\sqrt{2+\sqrt{2}}}{}$	$-{\sqrt {2}}+1$	$-{\sqrt {2}}-1$	$-{\sqrt {4-2{\sqrt {2}}}}$	${\sqrt {4+2{\sqrt {2}}}}$	$-202.5^{\circ }$	$-{\frac {9\pi }{8}}$
$165^{\circ }$	${\frac {11\pi }{12}}$	${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	$-{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	$-2+{\sqrt {3}}$	$-2-{\sqrt {3}}$	$-{\sqrt {6}}+{\sqrt {2}}$	${\sqrt {6}}+{\sqrt {2}}$	$-195^{\circ }$	$-{\frac {13\pi }{12}}$
$180^{\circ }$	$\pi$	$0$	$-1$	$0$	未 定义	$-1$	未 定义	$-180^{\circ }$	$-\pi$
$195^{\circ }$	${\frac {13\pi }{12}}$	$-{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	$-{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	$2-{\sqrt {3}}$	$2+{\sqrt {3}}$	$-{\sqrt {6}}+{\sqrt {2}}$	$-{\sqrt {6}}-{\sqrt {2}}$	$-165^{\circ }$	$-{\frac {11\pi }{12}}$
$202.5^{\circ }$	${\frac {9\pi }{8}}$	$-{\frac {\sqrt {2-{\sqrt {2}}}}{2}}$	$-\frac{\sqrt{2+\sqrt{2}}}{}$	${\sqrt {2}}-1$	${\sqrt {2}}+1$	$-{\sqrt {4-2{\sqrt {2}}}}$	$-{\sqrt {4+2{\sqrt {2}}}}$	$-157.5^{\circ }$	$-{\frac {7\pi }{8}}$
$210^{\circ }$	${\frac {7\pi }{6}}$	$-{\frac {1}{2}}$	$-{\frac {\sqrt {3}}{2}}$	${\frac {1}{\sqrt {3}}}$	${\sqrt {3}}$	$-{\frac {2}{\sqrt {3}}}$	$-2$	$-150^{\circ }$	$-{\frac {5\pi }{6}}$
$225^{\circ }$	${\frac {5\pi }{4}}$	$-{\frac {1}{\sqrt {2}}}$		$1$		$-{\sqrt {2}}$		$-135^{\circ }$	$-{\frac {3\pi }{4}}$
$240^{\circ }$	${\frac {4\pi }{3}}$	$-{\frac {\sqrt {3}}{2}}$	$-{\frac {1}{2}}$	${\sqrt {3}}$	${\frac {1}{\sqrt {3}}}$	$-2$	$-{\frac {2}{\sqrt {3}}}$	$-120^{\circ }$	$-{\frac {2\pi }{3}}$
$247.5^{\circ }$	${\frac {11\pi }{8}}$	$-\frac{\sqrt{2+\sqrt{2}}}{}$	$-{\frac {\sqrt {2-{\sqrt {2}}}}{2}}$	${\sqrt {2}}+1$	${\sqrt {2}}-1$	$-{\sqrt {4+2{\sqrt {2}}}}$	$-{\sqrt {4-2{\sqrt {2}}}}$	$-112.5^{\circ }$	$-{\frac {5\pi }{8}}$
$255^{\circ }$	${\frac {17\pi }{12}}$	$-{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	$-{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	$2+{\sqrt {3}}$	$2-{\sqrt {3}}$	$-{\sqrt {6}}-{\sqrt {2}}$	$-{\sqrt {6}}+{\sqrt {2}}$	$-105^{\circ }$	$-{\frac {7\pi }{12}}$
$270^{\circ }$	${\frac {3\pi }{2}}$	$-1$	$0$	未 定义	$0$	未 定义	$-1$	$-90^{\circ }$	$-{\frac {\pi }{2}}$
$285^{\circ }$	${\frac {19\pi }{12}}$	$-{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	$-2-{\sqrt {3}}$	$-2+{\sqrt {3}}$	${\sqrt {6}}+{\sqrt {2}}$	$-{\sqrt {6}}+{\sqrt {2}}$	$-75^{\circ }$	$-{\frac {5\pi }{12}}$
$292.5^{\circ }$	${\frac {11\pi }{8}}$	$-\frac{\sqrt{2+\sqrt{2}}}{}$	${\frac {\sqrt {2-{\sqrt {2}}}}{2}}$	$-{\sqrt {2}}-1$	$-{\sqrt {2}}+1$	${\sqrt {4+2{\sqrt {2}}}}$	$-{\sqrt {4-2{\sqrt {2}}}}$	$-67.5^{\circ }$	$-{\frac {3\pi }{8}}$
$300^{\circ }$	${\frac {5\pi }{3}}$	$-{\frac {\sqrt {3}}{2}}$	${\frac {1}{2}}$	$-{\sqrt {3}}$	$-{\frac {1}{\sqrt {3}}}$	$2$	$-{\frac {2}{\sqrt {3}}}$	$-60^{\circ }$	$-{\frac {\pi }{3}}$
$315^{\circ }$	${\frac {7\pi }{4}}$	$-{\frac {1}{\sqrt {2}}}$	${\frac {1}{\sqrt {2}}}$	$-1$		${\sqrt {2}}$	$-{\sqrt {2}}$	$-45^{\circ }$	$-{\frac {\pi }{4}}$
$330^{\circ }$	${\frac {11\pi }{6}}$	$-{\frac {1}{2}}$	${\frac {\sqrt {3}}{2}}$	$-{\frac {1}{\sqrt {3}}}$	$-{\sqrt {3}}$	${\frac {2}{\sqrt {3}}}$	$-2$	$-30^{\circ }$	$-{\frac {\pi }{6}}$
$337.5^{\circ }$	${\frac {15\pi }{8}}$	$-{\frac {\sqrt {2-{\sqrt {2}}}}{2}}$	${\frac {\sqrt {2+{\sqrt {2}}}}{2}}$	$-{\sqrt {2}}+1$	$-{\sqrt {2}}-1$	${\sqrt {4-2{\sqrt {2}}}}$	$-{\sqrt {4+2{\sqrt {2}}}}$	$-22.5^{\circ }$	$-{\frac {\pi }{8}}$
$345^{\circ }$	${\frac {13\pi }{12}}$	$-{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	$-2+{\sqrt {3}}$	$-2-{\sqrt {3}}$	${\sqrt {6}}-{\sqrt {2}}$	$-{\sqrt {6}}-{\sqrt {2}}$	$-15^{\circ }$	$-{\frac {\pi }{12}}$
$360^{\circ }$	$2\pi$	$0$	$1$	$0$	未 定义	$1$	未 定义	$0^{\circ }$	$0$

请注意，对于某些 $x$ 的值，正切、余切、正割和余割函数是未定义的。这是因为这些函数定义为 ${\frac {\sin(x)}{\cos(x)}}$ 、 ${\frac {\cos(x)}{\sin(x)}}$ 、 ${\frac {1}{\cos(x)}}$ 和 ${\frac {1}{\sin(x)}}$ ，分别。由于如果表达式包含除以 0，则该表达式是未定义的，因此在角度量度为 0 的情况下，这些函数是未定义的（取决于三角函数，$x$ 的正弦或余弦）。例如，90º 的正切函数（$\pi /2\,$ 弧度）等效于 ${\frac {\sin \left({\frac {\pi }{2}}\right)}{\cos \left({\frac {\pi }{2}}\right)}}$ 或者 ${\frac {1}{0}}$ ，这是一个未定义的值。