## Trigonometry (11th Edition) Clone

$$\sin(270^\circ-\theta)=-\cos\theta$$
$$X=\sin(270^\circ-\theta)$$ According to sine difference identity: $$\sin(A-B)=\sin A\cos B-\cos A\sin B$$ Expand $X$: $$X=\sin270^\circ\cos\theta-\cos270^\circ\sin\theta$$ *About $\sin270^\circ$ and $\cos270^\circ$ $$\sin270^\circ=\sin(-90^\circ)\hspace{2cm}\cos270^\circ=\cos(-90^\circ)$$ From the Negative-Angle Identities: $$\sin(-\theta)=-\sin\theta\hspace{2cm}\cos(-\theta)=\cos\theta$$ So, $$\sin270^\circ=-\sin90^\circ=-1\hspace{2cm}\cos270^\circ=\cos90^\circ=0$$ We replace the values just being found back into $X$: $$X=(-1)\cos\theta-0\sin\theta$$ $$X=-\cos\theta$$ Overall, $$\sin(270^\circ-\theta)=-\cos\theta$$