solve csc^2x - cscx + 9 =11
csc ^ 2 ( x ) - csc ( x ) + 9 = 11 Subtract 11 to both sides
csc ^ 2 ( x ) - csc ( x ) + 9 - 11 = 11 - 11
csc ^ 2 ( x ) - csc ( x ) - 2 = 0
Substitute :
csc ( x ) = u
u ^ 2 - u - 2 = 0
The exact solutions are :
u = - 1
and
u = 2
u = csc ( x ) so :
csc ( x ) = - 1
and
csc ( x ) = 2
csc ( - pi / 2 ) = - 1
csc ( x ) is a periodic function with period 2 pi n
where n is an integer
Solutions :
x = 2 pi n - pi / 2
and
x = 2 pi n - pi / 2 + 2 pi =
2 pi n - pi / 2 + 4 pi / 2 =
2 pi n + 3 pi / 2
csc ( pi / 6 ) = 2
csc ( 5 pi / 6 ) = 2
csc ( x ) is a periodic function with period 2 pi n
where n is an integer
Solutions :
x = 2 pi n + pi / 6
and
x = 2 pi n + 5 pi / 6
Final solutions :
x = 2 pi n - pi / 2
x = 2 pi n + 3 pi / 2
x = 2 pi n + pi / 6
x = 2 pi n + 5 pi / 6
P.S
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u ^ 2 - u - 2 = 0
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u ^ 2 - u - 2 = 0
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To solve the equation csc^2x - cscx + 9 = 11, we can rearrange it as a quadratic equation by substituting y = cscx:
y^2 - y + 9 = 11
Now, we have a quadratic equation:
y^2 - y - 2 = 0
To solve this equation, we can use factoring or the quadratic formula.
Using factoring, we need to find two numbers that multiply to give -2 and add up to -1 (which is the coefficient of y). These numbers are -2 and 1:
(y - 2)(y + 1) = 0
Setting each factor equal to zero:
y - 2 = 0 or y + 1 = 0
Solving for y:
y = 2 or y = -1
Since we set y = cscx, we can substitute these values back into the original equation:
cscx = 2 or cscx = -1
To solve for x, we need to find the inverse of csc. Remember that csc(x) is equal to 1/sin(x).
1/sinx = 2
sinx = 1/2
x = arcsin(1/2)
x = π/6 + 2πn or x = 5π/6 + 2πn, where n is an integer.
Similarly,
1/sinx = -1
sinx = -1
x = arcsin(-1)
x = π + 2πn, where n is an integer.
Therefore, the solutions to the equation csc^2x - cscx + 9 = 11 are:
x = π/6 + 2πn or x = 5π/6 + 2πn or x = π + 2πn, where n is an integer.
To solve the equation csc^2x - cscx + 9 = 11, we can use the quadratic formula.
Let's rearrange the equation to have the quadratic term on the left side:
csc^2x - cscx + (9 - 11) = 0
csc^2x - cscx - 2 = 0
Now, let's treat cscx as a variable, say y:
y^2 - y - 2 = 0
Comparing this new equation with the quadratic formula, we can identify that a = 1, b = -1, and c = -2. The quadratic formula is given by:
y = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values, we have:
y = (-(-1) ± √((-1)^2 - 4(1)(-2))) / (2(1))
Simplifying further:
y = (1 ± √(1 + 8)) / 2
y = (1 ± √9) / 2
y = (1 ± 3) / 2
Now, let's substitute y back with cscx:
cscx = (1 + 3) / 2 or cscx = (1 - 3) / 2
cscx = 4/2 or cscx = -2/2
Simplifying further:
cscx = 2 or cscx = -1
To find the values of x, we need to consider the range of the csc function.
The range of the csc(x) function is (-∞, -1] U [1, +∞).
Therefore, cscx = 2 does not have any solutions in the given range.
However, cscx = -1 does have solutions. In the standard position, the angles that have a csc of -1 are x = π OR x = -π (use the unit circle to visualize this).
So, the solutions to the given equation are:
x = π or x = -π