Trigonometric Functions, Identities and Equations

Multiply by $\dfrac{\pi}{180}$: $135 \times \dfrac{\pi}{180} = \dfrac{3\pi}{4}$.

$s = r\theta = 5 \times 0.8 = 4$. The angle must be in radians for this formula.

Area $= \tfrac12 r^2\theta = \tfrac12(36)(0.5) = 9$.

Since $\sec^2\theta - 1 = \tan^2\theta$, the expression becomes $\dfrac{\tan^2\theta}{\tan\theta} = \tan\theta$.

$R = \sqrt{25 + 144} = 13$ and $\tan\alpha = \dfrac{12}{5}$, giving $\alpha = 67.4°$. The maximum value is therefore 13.

Using $\sin^2\theta = 1 - \cos^2\theta$ gives $2\cos^2\theta - 3\cos\theta + 1 = 0$, so $(2\cos\theta-1)(\cos\theta-1) = 0$. Then $\cos\theta = \tfrac12$ or $\cos\theta = 1$.