AHL 3.11—Relationships between trig functions

We need $\cos (\pi -\theta )=-\cos \theta$, the cosine symmetry identity.

$\cos \theta =\frac{7}{25}$, so $\cos (\pi -\theta )=-\frac{7}{25}$.

$\cos (\pi -\theta )=-\frac{7}{25}$.

$\frac{7}{25}$ forgets the sign change entirely, treating cosine like sine.

$-\frac{24}{25}$ mistakenly computes $-\sin \theta$ using Pythagoras instead of $-\cos \theta$.

$\frac{24}{25}$ computes $\sin \theta$ via Pythagoras and gives no sign change, combining two errors.

Only A correctly applies $\cos (\pi -\theta )=-\cos \theta$.

$\frac{2\pi }{3}=\pi -\frac{\pi }{3}$.

$\sin (\pi -\theta )=\sin \theta$ with $\theta =\frac{\pi }{3}$.

$\sin \left(\frac{2\pi }{3}\right)=\sin \left(\frac{\pi }{3}\right)$.

$-\sin (\pi /3)$ would apply a sign change that belongs to cosine or tangent, not sine.

$\cos (\pi /3)$ confuses the $\pi -\theta$ identity with a co-function identity.

$-\cos (\pi /3)$ combines both previous mistakes.

Only A matches the correct sine symmetry identity.