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Home » Therefore, \(\cos 300^\circ = \boxed\frac12\). - Groen Casting

Therefore, \(\cos 300^\circ = \boxed\frac12\). - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Question: Compute $\tan 75^\circ$ using angle addition identities.

📌 Solution: Using the identity:

📌 \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}

📌 Let $a = 45^\circ$, $b = 30^\circ$, so:

📌 \tan 75^\circ = \tan(45^\circ + 30^\circ) = \frac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ}

📌 = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}} = \frac{\frac{\sqrt{3} + 1}{\sqrt{3}}}{\frac{\sqrt{3} - 1}{\sqrt{3}}} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}

📌 Multiply numerator and denominator by $\sqrt{3} + 1$:

📌 = \frac{(\sqrt{3} + 1)^2}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{3 + 2\sqrt{3} + 1}{3 - 1} = \frac{4 + 2\sqrt{3}}{2} = 2 + \sqrt{3}

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