Determine the value of theta; if theta ∈ [0°;90°] 3.1.1 sin theta = 0,8 3.1.2 2cos (2theta-60°) = 0,531

$$\sin \theta = 0.8$$ $$\theta = \arcsin(0.8)$$ $$\theta \approx 53.1301^\circ$$ $$\theta = \boxed{53.13^\circ}$$

3.1.2 Determine the value of $\theta$ if $2 \cos(2\theta - 60^\circ) = 0.531$.

$$2 \cos(2\theta - 60^\circ) = 0.531$$ $$\cos(2\theta - 60^\circ) = \frac{0.531}{2}$$ $$\cos(2\theta - 60^\circ) = 0.2655$$

$$2\theta - 60^\circ = \arccos(0.2655)$$ $$2\theta - 60^\circ \approx 74.591^\circ$$

$$2\theta = 74.591^\circ + 60^\circ$$ $$2\theta = 134.591^\circ$$ $$\theta = \frac{134.591^\circ}{2}$$ $$\theta \approx 67.2955^\circ$$ $$\theta = \boxed{67.30^\circ}$$

Question 3.2
Given point $M(-3;-2)$.

3.2.1 Length of OM.

$$OM = \sqrt{(-3)^2 + (-2)^2}$$ $$OM = \sqrt{9 + 4}$$ $$OM = \boxed{\sqrt{13}}$$

3.2.2 Value of $\sqrt{2} \sin \theta + 3 \sec^2 \theta$.

$$\sin \theta = \frac{y}{r} = \frac{-2}{\sqrt{13}}$$ $$\cos \theta = \frac{x}{r} = \frac{-3}{\sqrt{13}}$$ $$\sec \theta = \frac{1}{\cos \theta} = \frac{r}{x} = \frac{\sqrt{13}}{-3}$$

$$\sqrt{2} \sin \theta + 3 \sec^2 \theta = \sqrt{2} \left(\frac{-2}{\sqrt{13}}\right) + 3 \left(\frac{\sqrt{13}}{-3}\right)^2$$ $$= \frac{-2\sqrt{2}}{\sqrt{13}} + 3 \left(\frac{13}{9}\right)$$ $$= \frac{-2\sqrt{2}}{\sqrt{13}} + \frac{13}{3}$$

To rationalize the first term:

$$= \frac{-2\sqrt{2} \cdot \sqrt{13}}{\sqrt{13} \cdot \sqrt{13}} + \frac{13}{3}$$ $$= \frac{-2\sqrt{26}}{13} + \frac{13}{3}$$

To combine the terms, find a common denominator (39).

$$= \frac{-2\sqrt{26} \cdot 3}{13 \cdot 3} + \frac{13 \cdot 13}{3 \cdot 13}$$ $$= \frac{-6\sqrt{26}}{39} + \frac{169}{39}$$ $$= \boxed{\frac{169 - 6\sqrt{26}}{39}}$$

Question 3.3 Evaluate without using a calculator: $\frac{\cos 45^\circ}{\frac{\sqrt{3}}{2} \sin 30^\circ}$.

$$\cos 45^\circ = \frac{\sqrt{2}}{2}$$ $$\sin 30^\circ = \frac{1}{2}$$

$$\frac{\cos 45^\circ}{\frac{\sqrt{3}}{2} \sin 30^\circ} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{3}}{2} \cdot \frac{1}{2}}$$ $$= \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{3}}{4}}$$

$$= \frac{\sqrt{2}}{2} \times \frac{4}{\sqrt{3}}$$ $$= \frac{4\sqrt{2}}{2\sqrt{3}}$$ $$= \frac{2\sqrt{2}}{\sqrt{3}}$$

$$= \frac{2\sqrt{2}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}$$ $$= \frac{2\sqrt{6}}{3}$$ $$= \boxed{\frac{2\sqrt{6}}{3}}$$