Graph tan(t)>0sin(t)<0

Problem

tan(t)>0,sin(t)<0

Solution

1. Analyze the first condition tan(t)>0 The tangent function is positive in the first quadrant (0<t<π/2 and the third quadrant (π<t<(3*π)/2.

2. Analyze the second condition sin(t)<0 The sine function is negative in the third quadrant (π<t<(3*π)/2 and the fourth quadrant ((3*π)/2<t<2*π.

3. Find the intersection of the two conditions. The only quadrant where both tan(t)>0 and sin(t)<0 are true is the third quadrant.

4. Identify the interval for the solution. In terms of radians within one rotation, the solution is the set of all t such that π<t<(3*π)/2

5. Generalize the solution for all rotations. Since the trigonometric functions are periodic, the solution set is the union of intervals (π+2*k*π,(3*π)/2+2*k*π) for any integer k

Final Answer

π+2*k*π<t<(3*π)/2+2*k*π,k∈ℤ

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