Solve csc^2 x = 5 algebraically over the interval 0 <= x <= 2pi. Round to the nearest hundredth.
My answer: csc x = sqrt 5
= 2.24
can't csc x= -sqrt5 also?
and, you haven't solve for the x values yet..
Then x = arccsc sqrt 5
= 0.46
x = arccsc -sqrt 5, which is -arcsc sqrt 5, or -0.46.
To solve the equation csc^2(x) = 5 algebraically, we can start by taking the square root of both sides to eliminate the square.
√(csc^2x) = √5
Remember that the square root of a quantity squared is just the absolute value of the quantity. So we have:
|csc x| = √5
Now we can isolate the csc x by taking the reciprocal of both sides:
1 / |csc x| = 1 / √5
To get rid of the absolute value, we can solve two separate cases:
Case 1: csc x > 0
In this case, we have:
1 / csc x = 1 / √5
Next, we can take the reciprocal of both sides again:
csc x = √5
Case 2: csc x < 0
In this case, we have:
1 / (-csc x) = 1 / √5
Taking the reciprocal of both sides:
-csc x = √5
Now we have two separate equations to solve for csc x.
Case 1: csc x = √5
To solve for x, we take the inverse cosecant (or arcsin) of both sides:
x = arcsin(1/√5)
Using a calculator, this gives us the value of approximately 0.463 radians or 26.57 degrees.
Case 2: -csc x = √5
To solve for x, we take the inverse cosecant (or arcsin) of both sides, remembering that -csc x = csc(π - x):
π - x = arcsin(1/√5)
x = π - arcsin(1/√5)
Using a calculator, this gives us the value of approximately 2.678 radians or 153.43 degrees.
Thus, the solutions for x in the interval 0 <= x <= 2π are approximately 0.46 radians and 2.68 radians (or 26.57 degrees and 153.43 degrees). When rounded to the nearest hundredth, the solutions are 0.46 and 2.68, respectively.