8sin^2(x)+6sin(x)-9=0 simplify
To simplify the given equation, let's first substitute a variable. Let's use "u" instead of "sin(x)". The equation then becomes:
8u^2 + 6u - 9 = 0
Now we can solve this quadratic equation for "u" using the quadratic formula:
u = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 8, b = 6, and c = -9. Plugging these values into the quadratic formula:
u = (-6 ± √(6^2 - 4 * 8 * -9)) / (2 * 8)
u = (-6 ± √(36 + 288)) / 16
u = (-6 ± √324) / 16
Since √324 = 18, we have:
u = (-6 ± 18) / 16
Now we have two possible values for "u":
1. u = (-6 + 18) / 16 = 12/16 = 3/4
2. u = (-6 - 18) / 16 = -24/16 = -3/2
Remembering that we substituted "u" for "sin(x)", we can write two equations:
1. sin(x) = 3/4
2. sin(x) = -3/2
However, the value of "sin(x)" cannot be greater than 1 or less than -1. Therefore, the equation sin(x) = -3/2 has no solutions.
Thus, the simplified equation is: sin(x) = 3/4.
To simplify the given equation 8sin^2(x) + 6sin(x) - 9 = 0, we can try to factorize it.
Let's substitute sin(x) with a variable, say y. Therefore, the equation becomes:
8y^2 + 6y - 9 = 0
Now, we can try to factorize this quadratic equation. However, it's not a straightforward factorization. We can use the quadratic formula to solve for y, and then substitute back sin(x) for y to find the values of x.
The quadratic formula is given by:
y = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 8, b = 6, and c = -9.
Substituting these values into the formula, we get:
y = (-6 ± √(6^2 - 4(8)(-9))) / (2(8))
= (-6 ± √(36 + 288)) / 16
= (-6 ± √324) / 16
= (-6 ± 18) / 16
Therefore, we have two possible solutions for y:
1. y = (-6 + 18) / 16 = 12 / 16 = 3 / 4
2. y = (-6 - 18) / 16 = -24 / 16 = -3 / 2
Now, we substitute back sin(x) for y:
1. sin(x) = 3/4
2. sin(x) = -3/2
However, sin(x) cannot be equal to -3/2, as the range of the sine function is -1 to 1. Therefore, the only valid solution is sin(x) = 3/4.
To find the values of x, we can use inverse trigonometric functions. Taking the arcsin of both sides, we have:
x = arcsin(3/4)
The arcsin function gives us the angle whose sine is equal to 3/4. Therefore, we can use a calculator to find the value of x.
using the quadratic formula,
sin(x) = 3/4 (1±√5)
= -2.4 or 0.927
Now just find the two values of x which make that work.