Find all solutions of the equation 3sin^2 x -7sin x +2 = 0 in the interval [0,2pi).
The answer is x_1= ?? and x_2= ?? with x_1<x_2.
3 z^2 - 7 z + 2 = 0
(3 z - 1)(z - 2) = 0
z = 1/3 or z = 2
so if sin x = 1/3
x = .34 radians or pi - .34 = 2.8 radians
sin x = 2 has no solutions
To find the solutions of the equation 3sin^2(x) - 7sin(x) + 2 = 0 in the interval [0, 2pi), we can solve it using the quadratic formula.
The quadratic formula states: for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In our case, we have the equation 3sin^2(x) - 7sin(x) + 2 = 0, which can be rewritten as:
3sin^2(x) - 6sin(x) - sin(x) + 2 = 0
Rearranging the terms, we have:
(3sin^2(x) - 6sin(x)) - (sin(x) - 2) = 0
Factoring out common terms, we get:
3sin(x)(sin(x) - 2) - 1(sin(x) - 2) = 0
Now, we can factor out (sin(x) - 2):
(3sin(x) - 1)(sin(x) - 2) = 0
Setting each factor equal to zero, we have two equations to solve:
3sin(x) - 1 = 0 (Equation 1)
sin(x) - 2 = 0 (Equation 2)
Solving Equation 1:
3sin(x) - 1 = 0
3sin(x) = 1
sin(x) = 1/3
Using inverse trigonometric functions, we find the solution in the interval [0, 2pi) as:
x1 = arcsin(1/3) ≈ 0.34 rad
Solving Equation 2:
sin(x) - 2 = 0
sin(x) = 2
Since the range of the sine function is [-1, 1], there are no solutions for sin(x) = 2 in the interval [0, 2pi).
Therefore, the only solution in the interval [0, 2pi) for the given equation is:
x1 = 0.34 rad
To solve the equation 3sin^2(x) - 7sin(x) + 2 = 0 in the interval [0, 2pi), we will use the quadratic formula.
First, let's rewrite the equation in terms of sin(x) as a quadratic equation:
3sin^2(x) - 7sin(x) + 2 = 0
To make it easier, let's introduce another variable: let y = sin(x). The equation becomes:
3y^2 - 7y + 2 = 0
Now we can solve this quadratic equation for y:
Using the quadratic formula, y = (-b ± √(b^2 - 4ac)) / (2a), where a = 3, b = -7, and c = 2.
y = (-(-7) ± √((-7)^2 - 4(3)(2))) / (2(3))
y = (7 ± √(49 - 24)) / 6
y = (7 ± √25) / 6
y = (7 ± 5) / 6
We have two possibilities:
1. y = (7 + 5) / 6 = 12 / 6 = 2
2. y = (7 - 5) / 6 = 2 / 6 = 1/3
Now we can substitute y back into the equation y = sin(x):
1. sin(x) = 2
There are no solutions in the interval [0, 2pi) since the range of sin(x) is [-1, 1], and 2 is outside this range.
2. sin(x) = 1/3
To find the solutions for sin(x) = 1/3 in the interval [0, 2pi), we can use the inverse sine function or arcsin function. Let's call the solutions A and B:
A = arcsin(1/3)
B = pi - arcsin(1/3)
A is the angle in the first quadrant where sin(x) = 1/3, and B is the angle in the second quadrant.
Using a calculator, A ≈ 0.3398 radians or ≈ 19.47 degrees,
and B ≈ 2.802 radians or ≈ 160.53 degrees.
Now we convert the angles back to x:
x_1 = A = 0.3398 radians (approximately)
x_2 = B = 2.802 radians (approximately)
Therefore, the solutions in the interval [0, 2pi) are:
x_1 ≈ 0.3398 radians (approximately)
x_2 ≈ 2.802 radians (approximately)