Graph cos(theta)<0

Problem

cos(θ)<0

Solution

1. Identify the interval for the angle θ In a standard unit circle or a single period of the cosine function, we look at the interval [0,2*π]

2. Determine where the cosine function is negative. The cosine of an angle represents the xcoordinate on the unit circle. The xcoordinate is negative in the second and third quadrants.

3. Find the boundary points. The cosine function is zero at θ=π/2 and θ=(3*π)/2

4. Express the solution as an inequality. Since we are looking for values strictly less than zero, the function is negative when θ is between π/2 and (3*π)/2

5. Generalize the solution for all real numbers. Because cosine is periodic with a period of 2*π we add 2*π*k to the boundaries, where k is any integer.

Final Answer

π/2+2*π*k<θ<(3*π)/2+2*π*k,k∈ℤ

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