inequality question (1 Viewer)

[1zephyr]

How do i solve this inequality?

 

[InteGrand]

View attachment 32635

How do i solve this inequality?

Sketch the graph of y = sin x between x = 0 and x = 2π. Also sketch in the horizontal lines y = √(2)/2 (dotted line since we want a strict inequality for this one) and y = -½. We want the values of x such that the sine wave is between these two horizontal lines. Refer to this graph:

http://www.graphsketch.com/?eqn1_co..._lines=1&line_width=4&image_w=850&image_h=525

We basically need to find the x-values of the intersection points of the graph of y = sin x and those lines. This amounts to solving the equations sin x = √(2)/2 and sin x = -½ for 0 ≤ x ≤ 2π. You should know how to solve these, and going from left to right, you should get the x-values as being: π/4, 3π/4, 7π/6, 11π/6.

Then, we can see that the sine wave between 0 and 2π will be inside this band created by the horizontal lines (without being allowed to touch the upper line of y = √(2)/2) if and only if: 0 ≤ x < π/4 or 3π/4 < x ≤ 7π/6 or 11π/6 ≤ x ≤ 2π. So this is the answer.

 

[1zephyr]

Sketch the graph of y = sin x between x = 0 and x = 2π. Also sketch in the horizontal lines y = √(2)/2 (dotted line since we want a strict inequality for this one) and y = -½. We want the values of x such that the sine wave is between these two horizontal lines. Refer to this graph:

http://www.graphsketch.com/?eqn1_co..._lines=1&line_width=4&image_w=850&image_h=525

We basically need to find the x-values of the intersection points of the graph of y = sin x and those lines. This amounts to solving the equations sin x = √(2)/2 and sin x = -½ for 0 ≤ x ≤ 2π. You should know how to solve these, and going from left to right, you should get the x-values as being: π/4, 3π/4, 7π/6, 11π/6.

Then, we can see that the sine wave between 0 and 2π will be inside this band created by the horizontal lines (without being allowed to touch the upper line of y = √(2)/2) if and only if: 0 ≤ x < π/4 or 3π/4 < x ≤ 7π/6 or 11π/6 ≤ x ≤ 2π. So this is the answer.

lifesaver! cheers