Sekan

Sekan (lambang: sec; bahasa Inggris: secant) dalam matematika adalah perbandingan sisi miring segitiga dengan sisi yang terletak pada sudut (dengan catatan bahwa segitiga itu adalah segitiga siku-siku atau salah satu sudut segitiga itu 90^o). Perhatikan segitiga di kanan; berdasarkan definisi sekan di atas maka nilai sekan adalah

$\sec A={{\mbox{c}} \over {\mbox{b}}}\qquad \sec B={{\mbox{c}} \over {\mbox{a}}}$

Hubungan sekan dengan kosinus:

$\sec A={\frac {1}{\cos A}}\,$

Turunan dan Integral

Turunan

Turunan dari fungsi sec(x) adalah:

${\frac {d}{dx}}\sec {x}=\sec {x}\tan {x}$

Integral

Integral dari sec(x) bisa ditemukan melalui metode substitusi:

${\begin{aligned}\int \sec {x}\ dx&=\int \sec {x}\left({\frac {\sec {x}+\tan {x}}{\sec {x}+\tan {x}}}\right)\ dx\\&=\int {\frac {\sec ^{2}{x}+\sec {x}\tan {x}}{\sec {x}+\tan {x}}}\ dx\\\end{aligned}}$

$t=\sec {x}+\tan {x}\implies dt=\sec {x}\tan {x}+\sec ^{2}{x}$

${\begin{aligned}\int \sec {x}\ dx&=\int {\frac {1}{t}}\ dt\\&=\ln |t|+C\\&=\ln |\sec {x}+\tan {x}|+C\end{aligned}}$

Nilai sekan sudut istimewa

	Sinus	Kosinus	Tangen	Kotangen	Sekan	Kosekan
0°	$0$	$1$	$0$	$\infty$	$1$	$\infty$
8°	${\frac {\sqrt {2}}{10}}$	${\frac {7{\sqrt {2}}}{10}}$	${\frac {1}{7}}$	$7$	${\frac {5{\sqrt {2}}}{7}}$	$5{\sqrt {2}}$
15°	${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	$2-{\sqrt {3}}$	$2+{\sqrt {3}}$	${\sqrt {6}}-{\sqrt {2}}$	${\sqrt {6}}+{\sqrt {2}}$
16°	${\frac {7}{25}}$	${\frac {24}{25}}$	${\frac {7}{24}}$	${\frac {24}{7}}$	${\frac {25}{24}}$	${\frac {25}{7}}$
18°	${\frac {-1+{\sqrt {5}}}{4}}$	${\frac {\sqrt {10+2{\sqrt {5}}}}{4}}$	${\frac {(-5+3{\sqrt {5}}){\sqrt {10+2{\sqrt {5}}}}}{20}}$	${\frac {(1+{\sqrt {5}}){\sqrt {10+2{\sqrt {5}}}}}{4}}$	${\frac {(5-{\sqrt {5}}){\sqrt {10+2{\sqrt {5}}}}}{10}}$	$1+{\sqrt {5}}$
30°	${\frac {1}{2}}$	${\frac {\sqrt {3}}{2}}$	${\frac {\sqrt {3}}{3}}$	${\sqrt {3}}$	${\frac {2{\sqrt {3}}}{3}}$	$2$
36°	${\frac {\sqrt {10-2{\sqrt {5}}}}{4}}$	${\frac {1+{\sqrt {5}}}{4}}$	${\frac {(-1+{\sqrt {5}}){\sqrt {10-2{\sqrt {5}}}}}{4}}$	${\frac {(5+3{\sqrt {5}}){\sqrt {10-2{\sqrt {5}}}}}{20}}$	$-1+{\sqrt {5}}$	${\frac {(5+{\sqrt {5}}){\sqrt {10-2{\sqrt {5}}}}}{10}}$
37°	${\frac {3}{5}}$	${\frac {4}{5}}$	${\frac {3}{4}}$	${\frac {4}{3}}$	${\frac {5}{4}}$	${\frac {5}{3}}$
45°	${\frac {\sqrt {2}}{2}}$	${\frac {\sqrt {2}}{2}}$	$1$	$1$	${\sqrt {2}}$	${\sqrt {2}}$
53°	${\frac {4}{5}}$	${\frac {3}{5}}$	${\frac {4}{3}}$	${\frac {3}{4}}$	${\frac {5}{3}}$	${\frac {5}{4}}$
54°	${\frac {1+{\sqrt {5}}}{4}}$	${\frac {\sqrt {10-2{\sqrt {5}}}}{4}}$	${\frac {(5+3{\sqrt {5}}){\sqrt {10-2{\sqrt {5}}}}}{20}}$	${\frac {(-1+{\sqrt {5}}){\sqrt {10-2{\sqrt {5}}}}}{4}}$	${\frac {(5+{\sqrt {5}}){\sqrt {10-2{\sqrt {5}}}}}{10}}$	$-1+{\sqrt {5}}$
60°	${\frac {\sqrt {3}}{2}}$	${\frac {1}{2}}$	${\sqrt {3}}$	${\frac {\sqrt {3}}{3}}$	$2$	${\frac {2{\sqrt {3}}}{3}}$
72°	${\frac {\sqrt {10+2{\sqrt {5}}}}{4}}$	${\frac {-1+{\sqrt {5}}}{4}}$	${\frac {(1+{\sqrt {5}}){\sqrt {10+2{\sqrt {5}}}}}{4}}$	${\frac {(-5+3{\sqrt {5}}){\sqrt {10+2{\sqrt {5}}}}}{20}}$	$1+{\sqrt {5}}$	${\frac {(5-{\sqrt {5}}){\sqrt {10+2{\sqrt {5}}}}}{10}}$
74°	${\frac {24}{25}}$	${\frac {7}{25}}$	${\frac {24}{7}}$	${\frac {7}{24}}$	${\frac {25}{7}}$	${\frac {25}{24}}$
75°	${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	$2+{\sqrt {3}}$	$2-{\sqrt {3}}$	${\sqrt {6}}+{\sqrt {2}}$	${\sqrt {6}}-{\sqrt {2}}$
82°	${\frac {7{\sqrt {2}}}{10}}$	${\frac {\sqrt {2}}{10}}$	$7$	${\frac {1}{7}}$	$5{\sqrt {2}}$	${\frac {5{\sqrt {2}}}{7}}$
90°	$1$	$0$	$\infty$	$0$	$\infty$	$1$

Lihat pula

• Trigonometri