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\( (5.5) \sin 3 x=\cos x \) (5.6) \( \sin x=\sin 43^{\circ}+\sin 17^{\circ} \)

Ask by Parry Curry. in South Africa
Mar 11,2025

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**(5.5) Solutions for \( \sin 3x = \cos x \):** \[ x = \frac{\pi}{8} + \frac{k\pi}{2} \quad \text{or} \quad x = \frac{\pi}{4} + k\pi,\quad k\in\mathbb{Z}. \] **(5.6) Solutions for \( \sin x = \sin 43^\circ + \sin 17^\circ \):** \[ x = 77^\circ + 360^\circ k \quad \text{or} \quad x = 103^\circ + 360^\circ k,\quad k\in\mathbb{Z}. \]

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The Deep Dive

Did you know that the equation \( (5.5) \sin 3x = \cos x \) can lead us to explore multiple angles and identities? By using the triple angle formula for sine, we can rewrite it as \( 3 \sin x - 4 \sin^3 x = \cos x \). This allows us to dive deep into trigonometric identities and possible solutions in the interval \( [0, 360^{\circ}] \) or any periodic extension! For the equation \( (5.6) \sin x = \sin 43^{\circ} + \sin 17^{\circ} \), don't forget to use the sine addition formula! You can simplify the right side to find \( \sin x = 2 \sin\left(\frac{43^{\circ} + 17^{\circ}}{2}\right) \cos\left(\frac{43^{\circ} - 17^{\circ}}{2}\right) \). This breaks down to finding specific angle values that could make your calculations much more manageable and fun!

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