Q.If $\sin\theta=\dfrac{3}{5}$ and $\theta$ is acute, find the values of $\sin2\theta$ and $\cos2\theta$.

Since $\theta$ is acute, both $\sin\theta$ and $\cos\theta$ are positive, so the Pythagorean identity fixes $\cos\theta$ without any sign ambiguity, ready for direct substitution into the double-angle formulae.

Step 1 — find $\cos\theta$. $\cos^2\theta=1-\sin^2\theta=1-\dfrac{9}{25}=\dfrac{16}{25}$, so $\cos\theta=\pm\dfrac45$; since $\theta$ is acute (first quadrant), $\cos\theta=+\dfrac45$.

Step 2 — find $\sin2\theta$. $\sin2\theta=2\sin\theta\cos\theta=2\cdot\dfrac35\cdot\dfrac45=\dfrac{24}{25}$.

Step 3 — find $\cos2\theta$. Using $\cos2\theta=1-2\sin^2\theta$: $\cos2\theta=1-2\cdot\dfrac{9}{25}=1-\dfrac{18}{25}=\dfrac{7}{25}$. …

Alternatively, use the third form $\cos2\theta=2\cos^2\theta-1=2\cdot\dfrac{16}{25}-1=\dfrac{32}{25}-1=\dfrac{7}{25}$ — a third independent su …

Taking $\cos\theta=-\dfrac45$ despite $\theta$ being stated as acute (first quadrant, where cosine must be positive); also mixing up which of the three equivalent forms of …