determine exact solutions for each equation in the interval xe(0,2pi).
a)3 csc^2x-4
To find the exact solutions for the equation 3csc^2x - 4 = 0 in the interval x E (0, 2pi), we first isolate the term csc^2x by adding 4 to both sides of the equation:
3csc^2x = 4.
Next, divide both sides by 3:
csc^2x = 4/3.
To solve for x, we need to take the inverse cosecant (or arcsine) of both sides:
arccsc^2x = arccsc(4/3).
The inverse cosecant function is also known as the arcsin function, so we can rewrite the equation as:
arcsin(1/cscx) = arcsin(4/3).
Now, recall that the inverse sine function only operates within the range -pi/2 to pi/2. Therefore, we need to adjust the range of our solution accordingly. The interval is given as x E (0, 2pi), so the appropriate range for the arcsin function would be (0, pi).
Therefore, we can rewrite the equation as:
arcsin(1/cscx) = arcsin(4/3), for x E (0, pi).
To find the solutions, we can evaluate both sides of the equation using a calculator or through the unit circle. Evaluating arccsc(4/3) yields:
arccsc(4/3) ≈ 1.0908 radians.
So, we have:
arcsin(1/cscx) = 1.0908, for x E (0, pi).
To find the solutions for x, we take the sine of both sides:
sin(arcsin(1/cscx)) = sin(1.0908).
Simplifying further, we get:
1/cscx = sin(1.0908).
Recall that csc(x) is the reciprocal of sin(x), so we can rewrite the equation as:
1/sinx = sin(1.0908).
Now, to solve for x, we take the arcsine of both sides:
arcsin(1/sinx) = arcsin(sin(1.0908)).
The arcsin and sin functions cancel out, leaving us with:
1/sinx = 1.0908.
To isolate sinx, we reciprocate both sides of the equation:
sinx = 1/1.0908.
Evaluating this expression yields:
sinx ≈ 0.9163.
So, the solutions for x in the interval (0, pi) are x ≈ arcsin(0.9163) and x ≈ pi - arcsin(0.9163).
Using a calculator, we find:
x ≈ 1.1827 and x ≈ 1.9589.
Therefore, the exact solutions for the equation 3csc^2x - 4 = 0 in the interval x E (0, 2pi) are x ≈ 1.1827 and x ≈ 1.9589.