If sin x 23 x in quadrant I then find without finding x sin

If sin x = 2/3, x in quadrant I, then find (without finding x) sin(2x) = cos(2x) = tan(2x) =

Solution

since x is in first quadrent

given sinx=2/3

cosx=sqrt(1-sin^2(x))=sqrt(1-4/9)=sqrt(5/9)=sqrt(5)/3

(a)sin(2x)=2sinx*cosx=2*(2/3)*(sqrt(5)/3)=(4 sqrt5)/9

(b)cos(2x)=1-2sin^2(x)=1-2*(4/9)=1-(8/9)=1/9

(c) tan(2x)=sin(2x)/cos(2x)= ((4 sqrt5)/9)/(1/9)=4 sqrt5