Question

Please help with number 12

Answer 1

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))`

Answer 2

Explanation:

sin(195º)= -√6+√2/4

cos(195º)=-√6-√2/4

tan(195º)=2-√3