Graph natural log of tan(x)

Problem

ln(tan(x))

Solution

1. Identify the domain of the function ƒ(x)=ln(tan(x)) The natural logarithm requires its argument to be strictly positive, so we must have tan(x)>0

2. Determine the intervals where tan(x)>0 This occurs in the first and third quadrants of the unit circle, specifically x∈(k*π,k*π+π/2) for any integer k

3. Locate the vertical asymptotes. As x→k*π+ tan(x)→0 so ln(tan(x))→−∞ As x→(k*π+π/2)− tan(x)→∞ so ln(tan(x))→∞

4. Find the x-intercepts by setting ƒ(x)=0 This happens when tan(x)=1 which occurs at x=π/4+k*π

5. Analyze the periodicity. Since tan(x) has a period of π the function ƒ(x)=ln(tan(x)) also has a period of π

6. Sketch the curve. Within one period, such as (0,π/2) the graph starts at −∞ near x=0 crosses the x-axis at x=π/4 and rises toward ∞ as x approaches π/2

Final Answer

y=ln(tan(x))

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