Solve 3-3 csc x = 17 for (theta) is less than or equal to x is less than 2(pi); give answer to the nearest 100th of a radian
To solve the equation 3 - 3 csc(x) = 17, we'll go through the following steps:
Step 1: Rearrange the equation
Start by moving the 3 to the other side of the equation:
-3 csc(x) = 17 - 3
Step 2: Simplify the equation
Combine the numbers on the right side:
-3 csc(x) = 14
Step 3: Isolate the csc(x)
To isolate csc(x), divide both sides of the equation by -3:
csc(x) = 14 / -3
Step 4: Take the reciprocal
To convert from csc(x) to sin(x), take the reciprocal of both sides of the equation:
sin(x) = -3 / 14
Step 5: Find the angle
To get the angle x, we need to find the arcsine of both sides of the equation:
x = arcsin(-3 / 14)
Step 6: Convert to radians
The arcsine function gives the angle in radians, but since we need the answers rounded to the nearest hundredth of a radian, there is no need to make any further calculations.
Therefore, the solution to the equation 3 - 3 csc(x) = 17 for x being between (theta) and 2(pi) is x ≈ arcsin(-3 / 14) radians (to the nearest hundredth).