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The basics, the laws and relationships, and roots of unity.

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Right triangle \(\triangle ABC\) has a right angle at \(B\), and \(\angle BAC = 30^{\circ} = \frac{\pi}{6}\). Find \[\sin\angle BAC + \cos\angle BCA.\]

In triangle \(\triangle ABC\), \(AB = 3, AC = 4\), and \(\angle BAC = \frac{\pi}{2}\). What is \(BC\)?

Let \(ABC\) be an equilateral triangle. Point \(D\) lies on \(BC\) so that \(\angle BAD = 15^{\circ}\). Find \(\frac{[BAD]}{[CAD]}\).

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