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1. \( \sin x \cos y-\cos \alpha \sin y \) \( 2 \cdot \operatorname{Cos} a \cdot \cos 12^{\circ}+\sin a \sin ^{\circ} 12^{\circ} \) \( 3 \operatorname{Cos} \theta \operatorname{Cos} 4 \theta-\operatorname{Sin} \theta \operatorname{Sin} 4 \theta \) \( +\sin 43^{\circ} \cos 33^{\circ}-\cos 43^{\circ} \sin 33^{\circ} \) \( =\operatorname{Cos} 14^{\circ} \sin 6^{\circ}+\operatorname{Cos} 6^{\circ} \operatorname{Sin} A^{\circ} \) - \( \operatorname{Sin} 6 x \sin 5 x+\operatorname{Cos} 6 \operatorname{Cos} 5 x \) 1. \( \sin \left(x-20^{\circ}\right)\left(x \operatorname{ses} 20^{\circ}+\cos \left(x-20^{\circ}\right) \sin 20\right. \)

Ask by Ingram Perkins. in South Africa
Mar 05,2025

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Here are the simplified forms of the trigonometric expressions: 1. \( \sin(x - y) \) 2. \( \cos(a - 12^{\circ}) \) 3. \( \cos(5\theta) \) 4. \( \sin(10^{\circ}) \) 5. \( \sin(20^{\circ}) \) 6. \( \frac{1}{2} [\cos(x) - \cos(11x)] \) 7. \( \sin(x - 20^{\circ}) \left( x \cos 20^{\circ} + \cos(x - 20^{\circ}) \sin 20^{\circ} \right) \)

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Bonus Knowledge

Did you know that the sine and cosine functions were first introduced to Western mathematics through translations of Arabic works during the Middle Ages? Originally described using chords in a circle, these trigonometric functions have evolved and become essential tools in everything from astronomy to engineering. So the next time you calculate the distance to a star or design a roller coaster, remember the ancient mathematicians who paved the way! When applying trigonometric identities, one common mistake is neglecting to convert degrees to radians when necessary. Always check if your calculator is set to the correct mode! Additionally, be careful with the signs; it's easy to forget that \(\sin(-x) = -\sin(x)\) and \(\cos(-x) = \cos(x)\). A good tip is to sketch unit circles or triangles to visualize the relationships. This way, you’ll avoid mishaps on your mathematical journey!

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