Nicholas H.

College student: Tutor for several years

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Pre-Calculus

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Question:

Solve the equation $$2\cos^2 3\theta = \cos 3\theta$$ for $$ 0\le \theta <2\pi$$.

Nicholas H.

Answer:

$$2\cos^2 3\theta = \cos 3\theta$$ gives $$2\cos^2 3\theta-\cos 3\theta=0$$, so $$(\cos 3\theta)(2\cos 3\theta -1)=0$$. Therefore $$\cos 3\theta=0$$ or $$\cos 3\theta=1/2$$, so either $$3\theta=\pi/2, 3\pi/2, 5\pi/2, 7\pi/2, 9\pi/2, 11\pi/2$$ or $$3\theta=\pi/3, 5\pi/3, 7\pi/3, 11\pi/3, 13\pi/3, 17\pi/3$$. Dividing by 3 to find $$\theta$$, we obtain the solutions $$\theta=\pi/6,\pi/2,5\pi/6,7\pi/6,3\pi/2,11\pi/6$$ and $$\theta=\pi/9,5\pi/9,7\pi/9,11\pi/9,13\pi/9,17\pi/9$$.

Calculus

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Question:

Prove that a infinite sequence .99999...= 1

Nicholas H.

Answer:

$$ .999... = \sum_{i=1}^{\infty} \frac9{10^i} = \sum_{i=1}^{\infty} \frac{10-1}{10^i} = \sum_{i=1}^{\infty} (\frac{10}{10^{i-1}} - \frac{1}{10^i}) = \frac{1}{10^0} = 1$$ This proves that .999... = 1

Economics

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Question:

How does the increase of minimum wage affect the economy?

Nicholas H.

Answer:

Recent evidence shows that any effects of raising the minimum wage would not be extremely significant, but by raising the minimum wage, it is known that the purchasing power of individuals will increase, and reducing employee turnover, which are both positive effects for the consumers.

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