Question

Find all solutions to the equation csc x(2sinx-Sqrt 2)=0

Answer 1

45° and 135°

Explanation:

Given the equation csc x(2sinx-Sqrt 2)=0

This means csc x = 0 and 2sinx-√2 = 0

If 2sinx-√2 = 0, we will have;

2sinx = √2

Dividing both sides of the equation by 2 we have;

2sinx/2 = √2/2

sinx = √2/2

x = arcsin√2/2

x = 45°

Since sin(theta) is also positive in the second quadrant and the angle there is 180-theta, therefore;

x = 180 - 45°

x = 135°

The required solutions are 45° and 135°

Note that csc x = 0 cannot be solved to get x hence, the equation will be ignored.

Answer 2

The solution would be like this for this specific problem:

x = Pi/4 + 2kPi and 3Pi/4+ 2kPi, where k is any positive integer

If the solution continued forever, any positive integer will create just one more rotation 2pi, 4pi, 6pi, so on and so forth, thus, giving all of them the same value.