# How do you use the reference angles to find sin210cos330-tan 135?

Feb 3, 2015

$\sin \left(210\right) = \sin \left(180 + 30\right) = - \sin \left(30\right) = - \frac{1}{2}$
$\cos \left(330\right) = \cos \left(360 - 30\right) = \cos \left(- 30\right) = \cos \left(30\right) = \frac{\sqrt{3}}{2}$
$\tan \left(135\right) = \sin \frac{180 - 45}{\cos} \left(180 - 45\right) = \sin \frac{45}{-} \cos \left(45\right) = \frac{\frac{\sqrt{2}}{2}}{- \frac{\sqrt{2}}{2}} = - 1$
The answer, therefore, is
$\left(- \frac{1}{2}\right) \cdot \left(\frac{\sqrt{3}}{2}\right) - \left(- 1\right) = 1 - \frac{\sqrt{3}}{4} = 0.567$ (approximately)

The simple trigonometric identities used in the above calculations are:
sin(x+π) = −sin(x)
sin(x−π) = −sin(x)
sin(π−x) = sin(x)
cos(x+π) = −cos(x)
cos(x−π) = −cos(x)
cos(π−x) = −cos(x)
These any many other useful properties of trigonometric functions are explained in details in the chapter on Trigonometry on a Web site Unizor - a free Web site dedicated to advanced math for high school students.