DoE/Feb. - March 2010 Mathematics/P2 \[ \begin{array}{c} 10 \\ \mathrm{NSC} \end{array} \] QUESTION 10 10.1 If \( \sin 36^{\circ} \cos 12^{\circ}=p \) and \( \cos 36^{\circ} \sin 12^{\circ}=q \), determine in terms of \( p \) and \( q \) the value of: 10.1.1 \( \sin 48^{\circ} \) \( 10.1 .2 \sin 24^{\circ} \) \( 10.1 .3 \quad \cos 24^{\circ} \) (3) (3) (3) 10.2 Show that \( \sin ^{2} 20^{\circ}+\sin ^{2} 40^{\circ}+\sin ^{2} 80^{\circ}=\frac{3}{2} \) (HINT: \( 40^{\circ}=60^{\circ}-20^{\circ} \) and \( 80^{\circ}=60^{\circ}+20^{\circ} \) ) (7) 10.3 10.3.1 Prove: \( \frac{\sin ^{4} x+\sin ^{2} x \cos ^{2} x}{1+\cos x}=1-\cos x \) 10.3.2 For which values of \( x \) is \( \frac{\sin ^{4} x+\sin ^{2} x \cos ^{2} x}{1+\cos x}=1-\cos x \) NOT true? (2) [22]
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Did you know that the sine and cosine functions are not just mathematical concepts, but they also have deep historical roots? Trigonometric functions were first systematically studied by ancient Greek mathematicians, especially in the context of astronomy. The term "sine" is derived from the Arabic word "jayb," which was a mistranslation, ultimately bridging cultures through mathematics! When it comes to real-world applications, trigonometric functions are indispensable in fields like engineering, physics, and computer graphics. For instance, architects use sine and cosine to calculate forces and angles in structures, while game developers rely on these functions to depict realistic movements and animations. So next time you see a building or a video game, remember that trigonometry plays a significant role behind the scenes!