QuestionJuly 15, 2025

29. What is the exact value of sin75^circ (sqrt (2)+sqrt (6))/(4) (sqrt (6)-sqrt (2))/(4) (-sqrt (6)-sqrt (2))/(4) (sqrt (2)-sqrt (6))/(4)

29. What is the exact value of sin75^circ (sqrt (2)+sqrt (6))/(4) (sqrt (6)-sqrt (2))/(4) (-sqrt (6)-sqrt (2))/(4) (sqrt (2)-sqrt (6))/(4)
29. What is the exact value of sin75^circ (sqrt (2)+sqrt (6))/(4) (sqrt (6)-sqrt (2))/(4) (-sqrt (6)-sqrt (2))/(4) (sqrt (2)-sqrt (6))/(4)

Solution
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Answer

\frac{\sqrt{6} + \sqrt{2}}{4} Explanation 1. Use angle addition formula 75^\circ = 45^\circ + 30^\circ. Use \sin(a+b) = \sin a \cos b + \cos a \sin b. 2. Calculate \sin 45^\circ and \cos 45^\circ \sin 45^\circ = \frac{\sqrt{2}}{2}, \cos 45^\circ = \frac{\sqrt{2}}{2}. 3. Calculate \sin 30^\circ and \cos 30^\circ \sin 30^\circ = \frac{1}{2}, \cos 30^\circ = \frac{\sqrt{3}}{2}. 4. Apply values to the formula \sin 75^\circ = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}. 5. Simplify the expression \sin 75^\circ = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}.

Explanation

1. Use angle addition formula<br /> $75^\circ = 45^\circ + 30^\circ$. Use $\sin(a+b) = \sin a \cos b + \cos a \sin b$.<br />2. Calculate $\sin 45^\circ$ and $\cos 45^\circ$<br /> $\sin 45^\circ = \frac{\sqrt{2}}{2}$, $\cos 45^\circ = \frac{\sqrt{2}}{2}$.<br />3. Calculate $\sin 30^\circ$ and $\cos 30^\circ$<br /> $\sin 30^\circ = \frac{1}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$.<br />4. Apply values to the formula<br /> $\sin 75^\circ = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$.<br />5. Simplify the expression<br /> $\sin 75^\circ = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}$.
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