Please help with number 12

Question

asked Feb 3, 2020 119k views

2 Answers

Nikita Kalugin · Answer 1 · 2020-02-04T09:15:09+0000

Recall the rules

$\sin(a-b)=\sin(a) \cos(b) - \cos(a) \sin(b)$

$\cos(a-b)=\sin(a) \sin(b) + \cos(a) \cos(b)$

Use the suggested difference:

$\sin(195)=\sin(225-30)=\sin(225) \cos(30) - \cos(225) \sin(30)$

$\cos(195)=\cos(225-30)=\sin(225) \sin(30) + \cos(225) \cos(30)$

Since 225 and 30 are known angles, we can plug the values:

$\sin(225)=\cos(225)=-(1)/(√(2)),\quad \sin(30)=(1)/(2),\quad \cos(30)=(√(3))/(2)$

The expressions become

$\sin(195)=-(1)/(√(2)) \cdot (√(3))/(2) - \left(-(1)/(√(2))\right) \cdot (1)/(2) = (1-√(3))/(2√(2))$

$\cos(195)=-(1)/(√(2)) \cdot (1)/(2) + \left(-(1)/(√(2))\right) (√(3))/(2)=(-1-√(3))/(2√(2))$

As usual, we just use the definition of the tangent:

$\tan(195)=(\sin(195))/(\cos(195))=((1-√(3))/(2√(2)))/((-1-√(3))/(2√(2)))=(1-√(3))/(-1-√(3))$