Answer
$sin(2x) = -24/25$
$cos(2x) = 7/25$
$tan(2x) = -24/7$
Work Step by Step
1. Find sin(x) and tan(x)
$$sin^2(x) + cos^2(x) = 1$$ $$sin(x) = \pm \sqrt{1 - cos^2(x)}$$
- Since $\frac{1}{sin(x)} =csc(x) \lt 0$, sin(x) must be negative.
$$sin(x) = -\sqrt{1 - cos^2(x)} = - \sqrt{1 - (4/5)^2}$$ $$sin(x) = -\sqrt{1-16/25} = -\sqrt{9/25} = -3/5$$ $$tan(x) = \frac{sin(x)}{cos(x)} = \frac{-3/5}{4/5} = -3/4$$
2. Use the formulas to calculate sin(2x), cos(2x).
$$sin(2x) = 2sin(x)cos(x) = 2(-3/5)(4/5) = -24/25$$ $$cos(2x) = cos^2(x) - sin^2(x) = (4/5)^2 - (-3/5)^2$$ $$cos(2x) = 16/25 - 9/25 = 7/25$$
3. Calculate tan(2x)
$$tan(2x) = \frac{sin(2x)}{cos(2x)} = \frac{-24/25}{7/25} = -24/7$$
