Question

Find sin(2x), cos(2x), and tan(2x) from the given information.

Answer 1

Since
`\cot(x)=(2)/(3)` and
`\cot^(2) x+1=\csc^(2) x`, we know that:

`\left((2)/(3) \right)^(2)+1=\csc^(2) x\\\\(13)/(9)=\csc^(2) x\\\\\csc x=(√(13))/(3)`

If
`\csc x=(√(13))/(3)`, this means that
`\sin x=(3)/(√(13))` and by the Pythagorean identity,

`\sin^(2) x+\cos^(2) x=1\\\left((3)/(√(13)) \right)^(2)+\cos^(2) x=1\\(9)/(13)+\cos^(2) x=1\\\cos^(2) x=\frac{4}13}\\\cos x=(2)/(√(13))`

• Using the double angle formula for sine,
  `\sin(2x)=2\left((3)/(√(13)) \right)\left((2)/(√(13)) \right)=\boxed{(12)/(13)}`
• Using the double angle formula for cosine,
  `\cos(2x)=1-2\left((3)/(√(13)) \right)^(2)=\boxed{-(5)/(13)}`
• So, since tan=sin/cos,
  `\tan (2x)=(\sin(2x))/(\cos(2x))=((12)/(13))/(-(5)/(13))=\boxed{-(12)/(5)}`