Question

Find the exact value of sin 2u and cos 2u using double angle formula. cos u= -2/3

Answer 1

`\sin2u=2\sin u\cos u`

We know that
`\cos u=-\frac23`, so we can find out what
`\sin u` is, but there are two possibilities. By the Pythagorean theorem,


`\sin^2u+\cos^2u=1\implies \sin u=\pm√(1-\left(-\frac23\right)^2)=\pm\frac{\sqrt5}3`

and so


`\sin2u=2\left(\pm\frac{\sqrt5}3\right)\left(-\frac23\right)=\pm\frac{4\sqrt5}9`

Next,


`\cos2u=\cos^2u-\sin^2u`

and since
`x^2\ge0` for all
`x`, we end up with exactly one value for
`\cos2u`:


`\cos2u=\left(-\frac23\right)^2-\left(\pm\frac{\sqrt5}3\right)^2=-\frac19`