Duplication with cos α = 1/3

Problem

Given that $\cos\alpha = \dfrac{1}{3}$ with $\alpha$ in the first quadrant, compute $\sin(2\alpha)$, $\cos(2\alpha)$ and $\tan(2\alpha)$.

Solution

Since $\alpha$ is in the first quadrant, $\sin\alpha > 0$, so from the fundamental identity: $\sin\alpha = \sqrt{1-\cos^2\alpha} = \sqrt{1-\frac{1}{9}} = \frac{2\sqrt{2}}{3}.$ We now apply the duplication formulae: $\sin(2\alpha) = 2\sin\alpha\cos\alpha = 2\cdot\frac{2\sqrt{2}}{3}\cdot\frac{1}{3} = \boxed{\dfrac{4\sqrt{2}}{9}},$ $\cos(2\alpha) = 2\cos^2\alpha - 1 = \frac{2}{9} - 1 = \boxed{-\dfrac{7}{9}},$ $\tan(2\alpha) = \frac{\sin(2\alpha)}{\cos(2\alpha)} = \frac{4\sqrt{2}/9}{-7/9} = \boxed{-\dfrac{4\sqrt{2}}{7}}.$

Links

Topics: Trigonometric formulae
Concepts: Duplication formulae
Functions: Cosine · Sine · Tangent
Methods: Duplication formulae
Skills: Calculating · Using formulae
Exercise types: Computing expressions