Question

What is the solution to the trigonometric inequality 2-3csc(x) >8 over the interval..

Answer 1

The solution to the inequality 2 - 3csc(x) > 8 involves isolating csc(x), converting it to a sine function, and using the unit circle to determine where sin(x) is greater than -1/2 within the given interval.

Step-by-step explanation:

The solution to the trigonometric inequality 2 - 3csc(x) > 8 is found by first isolating the cosecant function on one side of the inequality:

1. Subtract 2 from both sides to get -3csc(x) > 6.
2. Divide both sides by -3, remembering to reverse the inequality sign since we are dividing by a negative number, which gives csc(x) < -2.
3. Since csc(x) is the reciprocal of sin(x), we have sin(x) > -1/2. We can find the solution by locating where sine is greater than -1/2 on the unit circle within the given interval.

The unit circle approach is useful to determine the exact values of x where the inequality holds true. The solutions will fall within specific intervals based on the periodic nature of the sine function. The sine function is greater than -1/2 in specific quadrants where the y-values are above -1/2.

Answer 2

To solve the trigonometric inequality 2-3csc(x) >8 over the interval, isolate the variable x by subtracting 2 from both sides, dividing by -3, and inverting the inequality symbol. The solution is 210° < x < 330°.

Step-by-step explanation:

To solve the trigonometric inequality 2-3csc(x) >8 over the interval, we need to isolate the variable x.

Here are the steps to solve the inequality:

1. Subtract 2 from both sides of the inequality to get -3csc(x) > 6.
2. Divide both sides of the inequality by -3 to get csc(x) < -2.
3. Invert the inequality symbol since we are dividing by a negative number to get sin(x) > -1/2.
4. Find the reference angle by finding the angle in the first quadrant that has the same sine value. The reference angle for sin(x) > -1/2 is 30°.
5. Since sin(x) is positive in the second and third quadrants, we can write the solution as 180° + 30° < x < 360° - 30°. Simplifying, we get 210° < x < 330°.

The complete question is:What is the solution to the trigonometric inequality 2-3csc(x) >8 over the interval..