QuestionJuly 7, 2025

Solve the following equation for all values of 0leqslant Theta lt 2pi -5cos2Theta -2sinTheta -4=3sinTheta -9 Answer Attempt 2 out of 2

Solve the following equation for all values of 0leqslant Theta lt 2pi -5cos2Theta -2sinTheta -4=3sinTheta -9 Answer Attempt 2 out of 2
Solve the following equation for all values of 0leqslant Theta lt 2pi -5cos2Theta -2sinTheta -4=3sinTheta -9 Answer Attempt 2 out of 2

Solution
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Answer

\Theta = 0, \pi, \frac{\pi}{6}, \frac{5\pi}{6} Explanation 1. Simplify the equation Start with -5\cos(2\Theta) - 2\sin(\Theta) - 4 = 3\sin(\Theta) - 9. Move terms to one side: -5\cos(2\Theta) - 2\sin(\Theta) - 3\sin(\Theta) - 4 + 9 = 0. 2. Combine like terms Combine \sin(\Theta) terms: -5\cos(2\Theta) - 5\sin(\Theta) + 5 = 0. 3. Use double angle identity Use \cos(2\Theta) = 2\cos^2(\Theta) - 1: -5(2\cos^2(\Theta) - 1) - 5\sin(\Theta) + 5 = 0. 4. Simplify further Distribute and simplify: -10\cos^2(\Theta) + 5 - 5\sin(\Theta) + 5 = 0, which simplifies to -10\cos^2(\Theta) - 5\sin(\Theta) + 10 = 0. 5. Solve for trigonometric identities Substitute \cos^2(\Theta) = 1 - \sin^2(\Theta): -10(1 - \sin^2(\Theta)) - 5\sin(\Theta) + 10 = 0. 6. Simplify quadratic equation Simplify: -10 + 10\sin^2(\Theta) - 5\sin(\Theta) + 10 = 0, leading to 10\sin^2(\Theta) - 5\sin(\Theta) = 0. 7. Factor the equation Factor out common term: 5\sin(\Theta)(2\sin(\Theta) - 1) = 0. 8. Solve each factor Solve 5\sin(\Theta) = 0: \sin(\Theta) = 0. Solutions are \Theta = 0, \pi. Solve 2\sin(\Theta) - 1 = 0: \sin(\Theta) = \frac{1}{2}. Solutions are \Theta = \frac{\pi}{6}, \frac{5\pi}{6}.

Explanation

1. Simplify the equation<br /> Start with $-5\cos(2\Theta) - 2\sin(\Theta) - 4 = 3\sin(\Theta) - 9$. Move terms to one side: $-5\cos(2\Theta) - 2\sin(\Theta) - 3\sin(\Theta) - 4 + 9 = 0$.<br /><br />2. Combine like terms<br /> Combine $\sin(\Theta)$ terms: $-5\cos(2\Theta) - 5\sin(\Theta) + 5 = 0$.<br /><br />3. Use double angle identity<br /> Use $\cos(2\Theta) = 2\cos^2(\Theta) - 1$: $-5(2\cos^2(\Theta) - 1) - 5\sin(\Theta) + 5 = 0$.<br /><br />4. Simplify further<br /> Distribute and simplify: $-10\cos^2(\Theta) + 5 - 5\sin(\Theta) + 5 = 0$, which simplifies to $-10\cos^2(\Theta) - 5\sin(\Theta) + 10 = 0$.<br /><br />5. Solve for trigonometric identities<br /> Substitute $\cos^2(\Theta) = 1 - \sin^2(\Theta)$: $-10(1 - \sin^2(\Theta)) - 5\sin(\Theta) + 10 = 0$.<br /><br />6. Simplify quadratic equation<br /> Simplify: $-10 + 10\sin^2(\Theta) - 5\sin(\Theta) + 10 = 0$, leading to $10\sin^2(\Theta) - 5\sin(\Theta) = 0$.<br /><br />7. Factor the equation<br /> Factor out common term: $5\sin(\Theta)(2\sin(\Theta) - 1) = 0$.<br /><br />8. Solve each factor<br /> Solve $5\sin(\Theta) = 0$: $\sin(\Theta) = 0$. Solutions are $\Theta = 0, \pi$.<br /> Solve $2\sin(\Theta) - 1 = 0$: $\sin(\Theta) = \frac{1}{2}$. Solutions are $\Theta = \frac{\pi}{6}, \frac{5\pi}{6}$.
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