Answer
$sin(2x) = -24/25$
$cos(2x) = -7/25$
$tan(2x) = 24/7$
Work Step by Step
$$tan(2x) = \frac{2 \space tan(x)}{1 - tan ^2(x)} = \frac{2 (-4/3)}{1 -
(-4/3)^2 } $$ $$tan(2x) = \frac{-8/3}{1 - 16/9} = \frac{-8/3}{-7/9} = 24/7$$
- To find cos(x) and sin(x):
$$tan^2(x) + 1 = sec^2(x)$$ $$tan^2(x) + 1 = \frac{1}{ cos^2(x)} $$ $$cos^2(x) = \frac{1}{tan^2(x) + 1 }$$ $$cos(x) = \pm \sqrt{ \frac{1}{tan^2(x) + 1 }}$$
x in Quadrant II: cosine values are negative. $$cos(x) = - \sqrt{ \frac{1}{tan^2(x) + 1 }} =-\sqrt{ \frac{1}{16/9 + 1 }} $$ $$-\sqrt{ \frac{1}{25/9 }} = - \sqrt{\frac{9}{25}} = - 3/5$$
$$sin^2(x) + cos^2(x) = 1$$ $$sin^2(x) + (-3/5)^2 = 1$$ $$sin^2(x) + 9/25 = 1$$ $$sin^2 (x) = 1 - 9/25 = 16/25$$ $$sin(x) = 4/5$$
$$sin(2x) = 2sin(x)cos(x) = 2(4/5)(-3/5) = -24/25$$ $$cos(2x) = cos^2(x) - sin^2(x) = (-3/5)^2 - (4/5)^2 $$ $$cos(2x) = 9/25 - 16/25= -7/25$$
