Question

what is the value of sec 165 degree​

Answer 1

Given:

sec(165°)

To find:

The value of sec(165°)

Solution:

sec(165°) = sec(120° + 45°)

Formula: sec(x) = 1 / cos(x)

$sec(120\° + 45\°) = \frac{1}{cos(120\° + 45\°)}$

Formula: cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

$=\frac{1}{cos(120\°)cos(45\°)-sin(120\°)sin(45\°)} \\\\= \frac{1}{cos(180\°-60\°)cos(45\°)-sin(180\°-60\°)sin(45\°)}\\\\= \frac{1}{-cos(60\°)cos(45\°)-sin(60\°)sin(45\°)}\\\\=\frac{1}{(-\frac{1}{2})(\frac{1}{\sqrt{2} } ) - (\frac{\sqrt{3} }{2} )(\frac{1}{\sqrt{2} } ) } \\\\=\frac{1}{\frac{-1-\sqrt{3} }{2\sqrt{2} } }\\\\=\frac{2\sqrt{2} }{-1-\sqrt{3} } \\\\=\frac{2\sqrt{2} }{-1-\sqrt{3} }*\frac{-1+\sqrt{3} }{-1+\sqrt{3} }$

Formula: (a + b)(a - b) = a² - b²

$=\frac{2\sqrt{2}(-1+\sqrt{3} )}{(-1)^{2} - (\sqrt{3} )^{2}}\\\\=\frac{2\sqrt{2}(-1 + \sqrt{3} ) }{1 - 3}\\\\= \frac{2\sqrt{2}(-1 + \sqrt{3} ) }{-2}$

= -√2(-1 + √3)

= √2 - √6

sec(165°) = √2 - √6

Therefore, sec(165°) = √2 - √6