Question

How do you verify $\cos \left(180°\right)=-1$?

Answer 1

Verify $\cos \left(180°\right)=-1$

$180°$ can be rewritten as $180°=\left(90°+90°\right)$.

Now, $\mathbf{cos}\mathbf{(}\mathbf{A}\mathbf{+}\mathbf{B}\mathbf{)}\mathbf{=}\mathbf{cos}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{·}\mathbf{cos}\mathbf{\left(}\mathbf{B}\mathbf{\right)}\mathbf{-}\mathbf{sin}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{·}\mathbf{sin}\mathbf{\left(}\mathbf{B}\mathbf{\right)}$.

Thus, $\cos \left(180°\right)=\cos \left(90°+90°\right)$.

$$\begin{gathered} \Rightarrow \cos \left(180°\right)=\cos \left(90°\right)·\cos \left(90°\right)-\sin \left(90°\right)·\sin \left(90°\right) \\ \Rightarrow \cos \left(180°\right)=\left(0\right)\times \left(0\right)-\left(1\right)\times \left(1\right) \\ \Rightarrow \cos \left(180°\right)=0-1 \\ \Rightarrow \cos \left(180°\right)=-1 \end{gathered}$$

Therefore, it is verified that $\cos \left(180°\right)=-1$.