Question

Find all solutions of the equation cos(x)sin(x) − 2cos(x) = 0. The answer is a b kπ where k is any integer and 0 < a < π.

a) (1/2, 2)
b) (0, 1)
c) (π/3, 1)
d) (π/2, 1)

Answer 1

The solution to the equation cos(x)sin(x) − 2cos(x) = 0 is found by factoring out cos(x) and considering the range 0 < x < π, leading to the solution ( π/2, 1), which corresponds to option d.

Step-by-step explanation:

The equation given is cos(x)sin(x) − 2cos(x) = 0. We can factor out the common term cos(x), which gives us cos(x)(sin(x) - 2) = 0. This implies either cos(x) = 0 or sin(x) - 2 = 0. Since sin(x) can never reach 2, we only consider cos(x) = 0.

The solutions to cos(x) = 0 are at x = (2k+1)π/2 where k is any integer, and this must be within the range 0 < x < π. Considering this range, the only solutions are x = π/2 and x = 3π/2. However, since 3π/2 is not between 0 and π, we discard it. Thus, the solution is (π/2, 1) matching option d.