### trigonometry

#### estimate angle between two lines y = 1000x and y = 999x

How to estimate the angle between line y = 1000 x and y = 999 x? I use the calculator and get 10^(-6) but how to approximate it by hand. Does it relate to Taylor Expansion?

The slope of the first line is 1000 and that of the second is 999. Simply use the formula \theta = \arctan\frac{m_1-m_2}{1+m_1m_2}, where m_1 is the slope of 1000 and m_2 is the slope of 999. You should get an answer of 1.00099×10^{-6} radians. To use a Taylor-McLaurin series, you should go by https://math.stackexchange.com/questions/348711/maclaurin-series-for-arctanx-by-successive-differentiation, and substitute the value of \frac{m_1-m_2}{1+m_1m_2} for x in each of the successive terms.

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