Question

Given: tan(180+a) = -3, find sin(4a), without using a calculator.

Answer 1

So far I have that tan(180+a) = tan(a), so I have tan(a) = -3

Answer 2

do you know sin(2a) = 2 sina cosa ?

Answer 3

yes, i applied that twice and got \[\sin(2a) = 2\sin(2a)\cos(2a)=2(2\sin(a)\cos(a))(\cos(2a))=4\sin(a)\cos(a)(\cos(2a))\]

Answer 4

sorry that first term should be sin(4a)

Answer 5

\[ \sin(4a)= 4\sin(a)\cos(a)(\cos(2a)) \]
now use cos(2a) = cos^2 a - sin^2 a

Answer 6

ok, \[\sin(4a)=4\sin(a)\cos(a)(\cos^2(a)-\sin^2(a))\] after expansion:\[4\sin(a)\cos(a)(\cos^2(a)-\sin^2(a))=4\sin(a)\cos^3(a)-4\sin^3(a)\cos(a)\]

Answer 7

the next step is draw a picture