1. Probleem 3.1: Bereken die waarde van $\sin^2 27^\circ + 2 \cos 54^\circ$. Formule: Gebruik die definisies van sinus en kosinus en die feit dat $\sin^2 x = (\sin x)^2$. Berekening: $\sin 27^\circ \approx 0.4540$, dus $\sin^2 27^\circ = (0.4540)^2 = 0.2061$. $\cos 54^\circ \approx 0.5878$, dus $2 \cos 54^\circ = 2 \times 0.5878 = 1.1756$. Som: $0.2061 + 1.1756 = 1.3817$. Antwoord: $\sin^2 27^\circ + 2 \cos 54^\circ \approx 1.3817$. 2. Probleem 3.2: Bereken $\frac{\sin 30^\circ \cdot \cot 45^\circ}{\cos 30^\circ \cdot \tan 60^\circ}$ sonder sakrekenaar. Formule en waardes: $\sin 30^\circ = \frac{1}{2}$, $\cot 45^\circ = 1$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$, $\tan 60^\circ = \sqrt{3}$. Berekening: $$\frac{\frac{1}{2} \times 1}{\frac{\sqrt{3}}{2} \times \sqrt{3}} = \frac{\frac{1}{2}}{\frac{\sqrt{3} \times \sqrt{3}}{2}} = \frac{\frac{1}{2}}{\frac{3}{2}} = \frac{1}{2} \times \frac{2}{3} = \frac{1}{3}$$ Antwoord: $\frac{1}{3}$. 3. Probleem 3.3: Gegee $\cos \theta = \frac{5}{13}$ en $\sin \theta < 0$. 3.3.1 Bereken $\sin \theta$. Formule: $\sin^2 \theta + \cos^2 \theta = 1$. Berekening: $$\sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{144}{169}$$ Dus $\sin \theta = -\sqrt{\frac{144}{169}} = -\frac{12}{13}$ (negatief omdat $\sin \theta < 0$). 3.3.2 Bereken $\sec \theta + \tan \theta + 1$. Formules: $\sec \theta = \frac{1}{\cos \theta} = \frac{13}{5}$. $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{12}{13}}{\frac{5}{13}} = -\frac{12}{5}$. Som: $$\sec \theta + \tan \theta + 1 = \frac{13}{5} - \frac{12}{5} + 1 = \frac{1}{5} + 1 = \frac{6}{5}$$ Antwoord: $\frac{6}{5}$. 4. Probleem 3.4: Los op vir $x$. 4.1 $5 \cos x + 2 = 4$ Isolasie: $$5 \cos x = 2 \Rightarrow \cos x = \frac{2}{5} = 0.4$$ Los op: $$x = \cos^{-1}(0.4)$$ Antwoord: $x \approx 66.42^\circ$ of $x \approx 360^\circ - 66.42^\circ = 293.58^\circ$ (in $0^\circ$ tot $360^\circ$). 4.2 $\frac{\csc x}{2} = -3$ Isolasie: $$\csc x = -6$$ Omdat $\csc x = \frac{1}{\sin x}$, dan: $$\sin x = \frac{1}{-6} = -\frac{1}{6}$$ Los op: $$x = \sin^{-1}(-\frac{1}{6})$$ Antwoord: $x \approx -9.59^\circ$ of $x \approx 180^\circ + 9.59^\circ = 189.59^\circ$ (in $0^\circ$ tot $360^\circ$). 5. Probleem 3.5: Gegee $\cos 20^\circ = k$ met $0 < k < 1$, bepaal $\tan 70^\circ$ met 'n skets. Gebruik die identiteit: $$\tan 70^\circ = \tan (90^\circ - 20^\circ) = \cot 20^\circ = \frac{\cos 20^\circ}{\sin 20^\circ} = \frac{k}{\sin 20^\circ}$$ Met 'n skets sien ons dat $\tan 70^\circ$ is die verhouding van die teenoorstaande teenoor die aanliggende sy in 'n regte driehoek met hoek $20^\circ$. Antwoord: $\tan 70^\circ = \cot 20^\circ = \frac{\cos 20^\circ}{\sin 20^\circ}$.