2 csc x - 3 = - 5
2 csc ( x ) - 3 = - 5 Add 3 to both sides
2 csc ( x ) - 3 + 3 = - 5 + 3
2 csc ( x ) = - 2 Divide both sides by 2
csc ( x ) = - 1
Solutions:
x = 2 pi n - pi / 2
and
x = 2 pi n + 3 pi / 2
n is an integer
To solve the equation 2 csc(x) - 3 = -5, we need to isolate the csc(x) term on one side. Here are the steps to solve it:
Step 1: Add 3 to both sides of the equation:
2 csc(x) - 3 + 3 = -5 + 3
2 csc(x) = -2
Step 2: Divide both sides of the equation by 2:
(2 csc(x))/2 = -2/2
csc(x) = -1
Now, we have csc(x) = -1.
To find the values of x that satisfy this equation, we need to recall the definition of the cosecant function. The cosecant of an angle is the reciprocal of the sine of that angle. So, we can rewrite the equation as:
1/sin(x) = -1
Step 3: By taking the reciprocal of both sides, we get:
sin(x) = -1
Now, we want to find the values of x in the range [0, 2π] that give us a sine of -1.
Step 4: From the unit circle or the graph of the sine function, we know that the sine function is equal to -1 at two points: π/2 and 3π/2.
Therefore, the values of x that satisfy the equation are x = π/2 and x = 3π/2.
In summary, the solutions to the equation 2 csc(x) - 3 = -5 are x = π/2 and x = 3π/2.
To solve the equation 2csc(x) - 3 = -5, you need to isolate the variable x.
Step 1: Add 3 to both sides of the equation:
2csc(x) - 3 + 3 = -5 + 3
2csc(x) = -2
Step 2: Divide both sides of the equation by 2:
(2csc(x))/2 = (-2)/2
csc(x) = -1
Step 3: Take the reciprocal of both sides of the equation (since csc(x) is the reciprocal of sin(x)):
1/csc(x) = 1/(-1)
sin(x) = -1
Step 4: Find the angle that has a sin value of -1 by referencing the unit circle. The angle that satisfies sin(x) = -1 is x = 270 degrees or x = (3π/2) radians, since sin(270 degrees) and sin(3π/2) both equal -1.
Therefore, the solution to the equation 2csc(x) - 3 = -5 is x = 270 degrees or x = (3π/2) radians.