If x is an acute angle and cosx=4/5, find the values of sin2x, sin3x, tan3x.
If $\displaystyle \cos{x} = \frac{4}{5}$, then $\displaystyle x = cos^{-1}\left( \frac{4}{5}\right) = 36.9^o$ (acute angle).
Simply substitute this x value into the other equations:
$\displaystyle \sin{(2 \times 36.9^o)} = $?
$\displaystyle \sin{(3 \times 36.9^o)} = $?
$\displaystyle \tan{(3 \times 36.9^o)} = $?
cos(x) = adjacent/hypotenuse.
So adjacent = 4
Hypotenuse = 5.
You can use pythagorus to work out the opposite side, and the use that to find sin(x) = opposite/hypotenuse.
$\displaystyle sin2x = 2sin(x)cos(x)$
$\displaystyle sin(3x) = sin(2x+x) = sin(2x)cos(x) + sin(x)cos(2x) $
$\displaystyle = 2sin(x)cos(x)cos(x)+sin(x)(cos^2(x)-sin^2(x)) $
$\displaystyle tan(3x) = \frac{sin(3x)}{cos(3x)} = \frac{sin(2x+x)}{cos(2x+x)}=\frac{sin(2x)cos(x) + sin(x)cos(2x)}{cos(2x)cos(x)-sin(2x)sin(x)} $
$\displaystyle =\frac{2sin(x)cos(x)cos(x)+sin(x)(cos^2(x)-sin^2(x))}{(cos^2(x)-sin^2(x))cos(x)-2sin(x)cos(x)sin(x))}$