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South Africa
Solve for \( x \) and \( y \) in each of the following sets of equation \( \begin{array}{ll}\text { (a) } y-2 x=2 & \text { and } x^{2}-2 x=3-y\end{array} \)
Algebra Feb 22, 2025
\( \begin{array}{lll}y-2 x=2 & \text { and } & x^{2}-2 x=3-y \\ y=-x-3 & \text { and } & y=2 x^{2}-3 x-3 \\ 3 x-y=2 & \text { and } & 3 y+9 x^{2}=4\end{array} \)
Algebra Feb 22, 2025
2.4. Use the cosine rule in \( \triangle \mathrm{POQ} \) to calculate the length of PQ in terms of \( \alpha \) and \( \beta \).
Trigonometry Feb 22, 2025
(8.5) \( \frac{3 \cos 150^{\circ} \cdot \sin 270^{\circ}}{\tan \left(-45^{\circ}\right) \cdot \cos 600^{\circ}} \) (8.6) \( \frac{\tan \left(180^{\circ}-x\right) \cdot \sin \left(90^{\circ}+x\right)}{\sin (-x)}-\sin y \cdot \cos \left(90^{\circ}-y\right) \) \( 8.7 \frac{\tan 30^{\circ} \cdot \sin 60^{\circ} \cdot \cos 25^{\circ}}{\cos 135^{\circ} \cdot \sin \left(-45^{\circ}\right) \cdot \sin 65^{\circ}} \) \( 8.8 \frac{\tan \left(180^{\circ}-x\right) \cdot \sin \left(90^{\circ}-x\right)}{\cos \left(90^{\circ}+x\right)}-\frac{\cos \left(180^{\circ}-x\right)}{\sin \left(90^{\circ}+x\right)} \) 8.9 \( \frac{\sin 189^{\circ}}{\tan 549^{\circ}}-\frac{\cos ^{2}\left(-9^{\circ}\right)}{\sin 99^{\circ}} \) Solving trigonometric equations (no calculator 1. If \( \sin \mathrm{A}=\frac{-3}{5} \) and \( 0^{\circ}<\mathrm{A}<270^{\circ} \) determine the value of: 1.1 \( \cos \mathrm{A} \) \( 1.2 \tan \mathrm{~A} \). (2.) If \( -5 \tan \theta-3=0 \) and \( \sin \theta<0 \), determine: \( 2.1 \sin ^{2} \theta^{\circ} \) \( 2.25 \cos \theta \) \( 2.31-\cos ^{2} \theta \). 3. If \( 13 \cos \theta+12=0 \) and \( 180^{\circ}<\theta<360^{\circ} \), evaluate: \( 3.1 \sin \theta \cos \theta \) \( 3.2 \tan \theta \) \( 3.3 \sin ^{2} \theta+\cos ^{2} \theta \). (4.) If \( 3 \tan \theta-2=0 \) and \( \theta \in\left[90^{\circ} ; 360^{\circ}\right] \), determine, the value of \( \sqrt{13}(\sin \theta- \)
Trigonometry Feb 22, 2025
8. Simplify without using a calculator. (8.1) \( \frac{\sin \left(180^{\circ}-x\right) \cdot \tan \left(360^{\circ}-x\right)}{\cos \left(80^{\circ}-x\right)} \times \frac{\cos \left(-180^{\circ}-x\right)}{\cos \left(360^{\circ}+x\right) \sin \left(360^{\circ}-x\right)} \) \( 8.2 \frac{\cos 135^{\circ} \sin 160^{\circ}}{\sin 225^{\circ} \cos 70^{\circ}} \) (8.3) \( \frac{\sin (-\theta)+\cos 120^{\circ}+\tan \left(-180^{\circ}-\theta\right)}{\sin ^{2} 225^{\circ}-\tan (-\theta)-\cos \left(90^{\circ}+\theta\right)} \) B.4 \( 4^{x} \frac{\sin 247^{\circ} \cdot \tan 23^{\circ} \cdot \cos 113^{\circ}}{\sin \left(-157^{\circ}\right)} \) (8.5) \( \frac{3 \cos 150^{\circ} \cdot \sin 270^{\circ}}{\tan \left(-45^{\circ}\right) \cdot \cos 600^{\circ}} \) 8.6) \( \frac{\tan \left(180^{\circ}-x\right) \cdot \sin \left(90^{\circ}+x\right)}{\sin (-x)}-\sin y \cdot \cos \left(90^{\circ}-y\right) \) \( 8.7 \frac{\tan 30^{\circ} \cdot \sin 60^{\circ} \cdot \cos 25^{\circ}}{\cos 135^{\circ} \cdot \sin \left(-45^{\circ}\right) \cdot \sin 65^{\circ}} \) 6.8) \( \frac{\tan \left(180^{\circ}-x\right) \cdot \sin \left(90^{\circ}-x\right)}{\cos \left(90^{\circ}+x\right)}-\frac{\cos \left(180^{\circ}-x\right)}{\sin \left(90^{\circ}+x\right)} \) \( 8.9 \frac{\sin 189^{\circ}}{\tan 549^{\circ}}-\frac{\cos ^{2}\left(-9^{\circ}\right)}{\sin 99^{\circ}} \) Solving trigonometric equations (no calculators) (1.) If \( \sin \mathrm{A}=\frac{-3}{5} \) and \( 0^{\circ}<\mathrm{A}<270^{\circ} \) determine the value of: \( 1.1 \cos A \) \( 1.2 \tan A \). (2.) If \( -5 \tan \theta-3=0 \) and \( \sin \theta<0 \), determine: \( 2.1 \sin ^{2} \theta^{\circ} \) \( 2.25 \cos \theta \) \( 2.3 \quad 1-\cos ^{2} \theta \) 3. If \( 13 \cos \theta+12=0 \) and \( 180^{\circ}<\theta<360^{\circ} \), evaluate: \( 3.2 \tan \theta \) \( 3.1 \sin \theta \cos \theta \) \( 3.3 \sin ^{2} \theta+\cos ^{2} \theta \). (4.) If \( 3 \tan \theta-2=0 \) and \( \theta \in\left[90^{\circ} ; 360^{\circ}\right] \), determine, the value of \( \sqrt{13}(\sin \theta-\cos \theta \) (5.) If \( \cos 52^{\circ}=k \) as illustrated in the diagram, determine each of the following i
Trigonometry Feb 22, 2025
3.2. Without using a calculator, simplify: \( \cos 20^{\circ} \cos 320^{\circ}-\sin 160^{\circ} \sin 40^{\circ} \)
Trigonometry Feb 22, 2025
3.1.3. \( \cos ^{2} 225^{\circ}-\sin ^{2} 225^{\circ} \)
Trigonometry Feb 22, 2025
4. If \( \pi<\theta<\frac{3}{3} \) with \( \cos (\theta)=-\frac{y \sqrt{3}}{5} \), find \( \sin (\theta) \)
Trigonometry Feb 22, 2025
\( \begin{array}{llll}\text { (c) } \sec \left(\frac{23 \pi}{12}\right) & \text { (f) } \csc \left(\frac{7 \pi}{4}\right) & \text { (g) } \cot \left(\frac{3 \pi}{4}\right) & \text { (h) } \sin \left(\frac{19 \pi}{12}\right) \\ \text { (i) } \tan \left(-\frac{11 \pi}{6}\right) & \text { (j) } \cot \left(\frac{13 \pi}{2}\right) & \text { (k) } \csc \left(-\frac{7 \pi}{4}\right) & \text { (l) } \cot (225)\end{array} \)
Trigonometry Feb 22, 2025
7. (a) A system of linear equations with fewer equations than unknowns is often underdetermined system. Suppose that such a system is consistent. Ext it must have an infinite number of solutions. (b) A system of linear equations with more equations than unknowns is off a overdetermined system. Can such a system be consistent? Give an exa system with three equations in two unknowns to illustrate your answer. 8. A system of linear equations is called homogeneous if the right hand side of e
Algebra Feb 22, 2025
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