Solve the System of @WORD tan(x)<0 , sin(x)<0

Problem

tan(x)<0,sin(x)<0

Solution

1. Identify the quadrants where sin(x)<0 The sine function is negative in the third quadrant (Q*3 and the fourth quadrant (Q*4.

2. Identify the quadrants where tan(x)<0 The tangent function is negative in the second quadrant (Q*2 and the fourth quadrant (Q*4.

3. Find the intersection of the two conditions. Both inequalities are satisfied only in the fourth quadrant (Q*4.

4. Express the solution for the fourth quadrant in terms of x In the standard interval [0,2*π) this corresponds to (3*π)/2<x<2*π

5. Generalize the solution by adding multiples of the period 2*π where n is any integer.

Final Answer

x∈((3*π)/2+2*n*π,2*π+2*n*π),n∈ℤ

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