9. The idenlity \( \sin \left(30^{\circ}+x\right)+\sin \left(30^{\circ}-x\right)=\cos x \) is given. Using this identity it can be shown Uiat sin43 \( +\sin 17^{\circ}=\cos 13^{\circ} \) without a calculator. (d) Use the above identity to delermine the value of \( \sin 40^{\circ}+\sin 20^{\circ} \)

To solve for \( \sin 40^{\circ} + \sin 20^{\circ} \) using the given identity \( \sin (30^{\circ} + x) + \sin (30^{\circ} - x) = \cos x \), we can recognize that we can set \( x = 10^{\circ} \). This gives us: \[ \sin(30^{\circ} + 10^{\circ}) + \sin(30^{\circ} - 10^{\circ}) = \cos(10^{\circ}). \] This simplifies to: \[ \sin(40^{\circ}) + \sin(20^{\circ}) = \cos(10^{\circ}). \] Thus, the value of \( \sin 40^{\circ} + \sin 20^{\circ} \) is \( \cos 10^{\circ} \). --- While it might seem a bit tricky to remember all these sine and cosine identities, they actually play a crucial role in various fields like engineering, physics, and computer graphics! Without them, modeling waves, oscillations, and even animations would be far more complex. And if you're eager to dive deeper into trigonometric identities, picking up a book on mathematical proofs or advanced trigonometry can be quite a treat! You’ll find all sorts of playful insights and nifty tricks that will make you the go-to math whiz among your friends!