Question

Please help with this problem

Answer 1

First of all, we have

`\sin^2(u)+\cos^2(u)=1 \implies \cos(u)=\pm√(1-\sin^2(u))`

Since u lies in the 4th quadrant, the cosine is positive, so we have

`\cos(u)=\sqrt{1-(9)/(25)}=(4)/(5)`

Now, the double angles formula: we have

`\sin(2u)=2\sin(u)\cos(u)=2\cdot\left(-(3)/(5)\right)\cdot (4)/(5)=-(24)/(25)`

`\cos(2u)=cos^2(u)-\sin^2(u)=(16)/(25)-(9)/(25)=(7)/(25)`

For the tangent, we can simply use the definition:

`tan(2u)=(\sin(2u))/(\cos(2u))=(-(24)/(25))/((7)/(25))=-(24)/(7)`