Angle x is acute and is such that tan x = sqrt(2)/4 (a) Show clearly that the exact value of sin x is 1/3 (b) Hence show that sin 2x = (4/9)sqrt(2)

Angle x is acute such that

`tan(x)=sqrt(2)/4. `

Thus we have a right triangle with legs of length sqrt(2) and 4. The hypotenuse of the triangle can be found using the Pythagorean theorem:

`h^2=(sqrt(2))^2+4^2`

`h^2=18`

`h=3sqrt(2)`

To find the sine, we take the ratio of the side opposite x to the hypotenuse:

`sin(x)=sqrt(2)/3sqrt(2)`

`=1/3`

Also, cos(x) is the ratio of the side adjacent to the hypotenuse so:

`cos(x)=4/(3sqrt(2))=2sqrt(2)/3`

The `sin(2A)=2sin(A)cos(A)` so:

`sin(2x)=2(1/3)((2sqrt(2))/3)`

`=2/3*2/3*sqrt(2)`

`=4/9*sqrt(2)`